Integral equations for the heterogeneous, Markovian SIS process with time-dependent rates in time-variant networks

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1 Integral equations for the heterogeneous, Markovian SIS process with time-dependent rates in time-variant networks P. Van Mieghem Delft University of Technology 2 April 208 Abstract We reformulate the governing equation for the nodal infection probability of the eact heterogeneous SIS process with time-dependent infection and curing rates on any network into two equivalent integral equations, that provide a di erent view on and a more natural description of the SIS dynamic process in temporal or time-variant networks. From the integral equations, we deduce general bounds. Introduction A huge number of papers has studied the SIS epidemic process on networks [], mainly because of three reasons. First, the SIS epidemic models a simpli ed version of a di usion process, which describes a kind of transport between nodes of its underlying graph [2]. Transport of items between nodes is major function of a network and is often approimated to rst order by an epidemic type of stochastic di usion, such as real viruses and diseases in a population, malware in digital communication networks and the spread of information (ideas, opinions, news, alarms, etc.) in social networks. Second and as earlier mentioned in [3], the SIS epidemic process is one of the simplest members of a particularly popular class of dynamic processes on networks called the Local rule-global emergent properties (LrGep) class, where the collective action of the local rules eecuted at each node gives rise to a comple, emergent global behavior. Some eamples of the LrGep-class are epidemic models (such as SIS and SIR) and more general reaction-di usion processes [], the Ising spin model [4], the Kuramoto coupled-oscillator model [5], sandpiles as models for self-organized criticality [6, 7, 8] and opinion models [9, 0]. Many LrGep models depend heavily on the underlying network topology and feature, in general, a phase transition []. The phase transition of the SIS and SIR epidemic models are similar to that of the Kuramoto-type of synchronization models, an observation that has received attention in human brain networks (see e.g. [2]). Third, we believe that the Markovian SIS epidemic model has the highest potential to analytically study that phase transition in networks. Pursuing the latter is still a worthwhile endeavor, because, so far, there does not eist (apart from asymptotic Faculty of Electrical Engineering, Mathematics and Computer Science, P.O Bo 503, 2600 GA Delft, The Netherlands; P.F.A.VanMieghem@tudelft.nl

2 studies [3]) an eact analysis, not even for the complete graph [4]. Before continuing, we remark that the phase transition in SIS and SIR epidemics occurs due to a change in the parameters (ratio of infection to curing rate) of the model, which might be di erent from a phase transition due to the non-linear dynamics of the process. Both the SIS and SIR Markovian model on any network are linear processes (as any Markov process [5]) and linear dynamic processes do not ehibit jumps, limit cycles nor chaotic behavior that may occur in some non-linear processes [6]. Here, the most general description of a heterogeneous Markovian SIS process with time-dependent rates on a time-variant network is recast into integral equations. Our main results are the derivations of integral equations (7) and (9) from the SIS governing di erential equation for the nodal infection probability, from which bounds are deduced in Section 4. 2 Markovian SIS governing equation The graph G of the network consists of the set N of N nodes and the set L of L links and is represented by an N N adjacency matri A with elements a ij = if there is a directed link from node i to j, else a ij = 0. The eact heterogeneous Markovian SIS governing equation [7, 5, 8] for the infection probability E [X i ] = Pr [X i = ] of node i is de [X i (t)] = E " i X i (t) + ( X i (t)) # ki a ki X k (t) where the Bernoulli random variable X i 2 f0; g de nes the two possible states, the infected state X i = and the healthy state X i = 0 of node i. The nodal curing is a Poisson process with rate i, while the infection from node k towards node i is also described by a Poisson process with rate ki. All Poisson processes are independent. When node i is infected at time t and X i (t) =, only the rst term in () on the right-hand side between the brackets [:] a ects and decreases with rate i the change in infection probability with time d Pr[X i(t)=] (left-hand side in ()). When node i by a rate P N k= kia ki X k (t) due to all its infected, direct neighbors. We de ne the nodal curing vector e = ( ; 2 ; : : : ; N ) and the weighted adjacency matri A e with element ea ij = ij a ij. Both the topology is healthy and X i (t) = 0, only the second term between the brackets [:] increases d Pr[X i(t)=] and the rates may dependent upon time; thus e A is generally a time-variant, asymmetric matri with zero elements on the diagonal (ea ii = 0 for all i N). An epidemic spreads along infectious links, which are network links with one end node infected and the other end node healthy. The probability that there is an infectious link directed from node i to j is Pr [X i = ; X j = 0] = E [X i ( X j )] = E [X i ] E [X i X j ] (2) which vanishes if i = j. With the de nition (2), the SIS governing equation () becomes de [X i (t)] + i (t) E [X i (t)] = k= () ea ki (t) Pr [X k (t) = ; X i (t) = 0] (3) k= These joint probabilities Pr [X k = ; X i = 0] in a Markovian SIS process can be determined precisely as shown in [7, 5], but require the knowledge of the joint probabilities over all triples of nodal 2

3 states, which in turn requires all combinations of four nodal states and so on. Eventually, the eact determination leads to 2 N linear di erential equations and this eponentially large number in N is a nger print of the NP-hard nature of SIS epidemics on networks. Using in (3) the conditional probability Pr [X k = ; X i = 0] = Pr [X k = jx i = 0] Pr [X i = 0] and E [X i ] = Pr [X i = 0], an alternative form of the heterogeneous Markovian SIS governing equation is de [X i (t)] + f i (t) + b i (t)g E [X i (t)] = b i (t) (4) where the overall infection rate of the healthy node i from all its neighbors is b i (t) = ea ki (t) Pr [X k (t) = jx i (t) = 0] (5) k= which will play a crucial role in the sequel. If de[x i(t)] = 0 at a time t = > 0, then the infection probability Pr [X i () = ] = E [X i ()] of node i is etremal and equal to Pr [X i () = ] = b i () i () + b i () (6) which resembles [9] the steady-state infection probability of node i in the N-Intertwined Mean-Field Approimation (NIMFA) [20], that led to a partial fraction epansion [2]. 3 An integral equation for Markovian SIS epidemics In appendi A, we reformulate a linear di erential equation of the type of the SIS di erential equation (4) into an integral equation. We apply (2) to the SIS di erential equation (4) and deduce the integral equation Pr [X i (t) = 0] = i (s) e s f i(u)+b i (u)gdu ds + Pr [X i () = 0] e f i(u)+b i (u)gdu connecting the healthy probability of node i at any two time points t and. The integral representation (7) may more naturally describe temporal networks [22], whose topology (and thus adjacency matri) changes over time. The complicating quantity in (7) is b i (t) de ned in (5), where the infectious treat ea ki (u) Pr [X k (u) = jx i (u) = 0] towards the healthy node i from its neighbor k, depends on the product of three independent factors: (7) the link eistence a ki (u) at time u which enables the transmission, the infection rate ki (u) at time u which characterizes the strength of the infectious activity and the likeliness Pr [X k (u) = jx i (u) = 0] of transmission over link k! i, because the SIS process only spreads over infectious links. The positive integrand in (7) shows that the probability that node i is healthy is always positive in a nite graph, provided that the curing rate i (s) is not zero at all times s 2 (0; t] and the infection rates ij are nite. When the time t increases while < t is ed and the curing rate i (u) i;min > 0, the last term containing the in uence of Pr [X i () = 0] at time on the actual healthy probability at time t disappears eponentially fast. The asymptotic time regime follows from (7) as lim Pr [X i (t) = 0] = lim t! t! i (s) e R s f i(u)+b i (u)gdu ds 3 e f i(u)+b i (u)gdu

4 Using de Hospital s rule and assuming that the limits lim t! i (t) = i; and lim t! ea ki (t) = ea ki; in (5) eist, we nd that lim Pr [X i (t) = 0] = t! i; i; + P N k= ea ki; lim t! Pr [X k (t) = jx i (t) = 0] which has the form (6) of an etremal probability. The governing equation () does not contain information to conclude that lim t! Pr [X k (t) = jx i (t) = 0] = 0, but, in any nite graph [23, 5], the SIS epidemic process ends in the absorbing, all-healthy state where lim t! Pr [X i (t) = 0] = for any node i 2 N. In other words, the infection has disappeared in the network after su ciently long time. The homogeneous case with ed rates ( ij (t) =, i (t) = and e ective infection rate = ) is the simplest version of the general epression (7), Pr [X i (t) = 0] = e (t s) e P N k= a ki s Pr[X k(u)=jx i (u)=0]du ds + Pr [X i () = 0] e (t ) e P N k= a ki Pr[X k(u)=jx i (u)=0]du (8) The homogeneous NIMFA reformulation, where v k (u) approimates the nodal infection probability Pr [X k (u) = ] by assuming independence such that v k (u) = Pr [X k (u) = ] = Pr [X k (u) = jx i (u) = 0], follows from (8) with v k (u) = v i (t) = e (+d i)(t v k (u) as s) e P N k= a ki s vk(u)du ds + v i () e (+d i)(t ) e P N k= a ki v k(u)du For an SIS Markovian process with ed rates, we know [24] that E [X i X j ] E [X i ] E [X j ], so that Pr [X k (u) = jx i (u) = 0] Pr [X k (u) = ]. Then, (8) shows that NIMFA lower bounds the healthy nodal probability, Pr [X i (t) = 0] v i (t), as found earlier [3]. 3. An alternative integral equation to (7) We demonstrate in Appendi B that the integral equation (7) is equivalent to ln Pr [Xi (t) = 0] = Pr [X i () = 0] i (u) Pr [X i (u) = ] Pr [X i (u) = 0] du A general integrated governing equation is presented in (28) in Appendi B. b i (u) du (9) By the mean-value theorem (see e.g. [5, Chapter 5]), there eist a point in time 2 [; t] for which (9) equals ln Pr[Xi (t)=0] Pr[X i ()=0] Pr [X i () = ] = i () t Pr [X i () = ] b i () (0) If the interval [; t] is not large, we may assume that Pr [X i () = ] at a particular (unknown) time is a good approimation for Pr [X i (u) = ] at any time point u 2 (; t). Thus, for a suitably chosen timestep h = t time approimation,, we may transform the continuous-time setting from (0) or from (27) into a discrete Pr [X i (t k+ ) = 0] = Pr [X i (t k ) = 0] e h i (t k ) Pr[X i (t k )=] Pr[X i (t k )=] 4 b i (t k )

5 where the (k + )-th time slot is epressed in terms of the k-th time slot, in which the infection probability Pr [X i (u) = ] = Pr [X i (t k ) = ] is constant for all u 2 [t k ; t k + h) in time slot k and t k+ = t k + h. Such approimation might be valuable for an actual disease spread over a timevariant contact network, especially when additional information about the conditional probability Pr [X k (t k ) = jx i (t k ) = 0] can be obtained. After solving (0) for Pr [X i () = ], we obtain Pr [X i () = ] = b i () + i () + b i () + Pr[X ln i (t)=0] Pr[X i ()=0] t ln Pr[X i (t)=0] Pr[X i ()=0] t and () resembles the etremal probability (6), but now complemented by the di erential quotient ln(pr[x i(t)=0]) ln(pr[x i ()=0]) t that involves the infection probabilities at the beginning time and ending time t of the interval [; t]. This generalized form () for a particular (unknown) time may suggest that each infection probability at time t obeys, to rst order, the etremal probability form Pr [X i (t) = ] = i (t) r i (t)+r i (t), in which r i (t) is a correction to the infectious link activity b i (t) = P N k= ea ki (t) Pr [X k (t) = jx i (t) = 0] at time t towards node i, when healthy, from all its infected neighbors. The SIS process possesses a quasi-stationary regime, also called the metastable regime. In the metastable regime, the nodal infection or healthy probability hardly changes anymore. Hence, if both time and t belong to the metastable regime, then Pr [X i (t) = 0] ' Pr [X i () = 0] ' Pr [X i (u) = 0] for u 2 [; t], so that () reduces to the equilibrium form given by the etremal probability (6). However, it is possible that Pr [X i (t) = 0] = Pr [X i () = 0] for a time and t in the transient regime as reported in [8], but where Pr [X i (u) = 0] for u 2 [; t] is not equal to Pr [X i (t) = 0] = Pr [X i () = 0] so that the etremal probability form () may not be a good approimation for those intermediate time points u 2 [; t]. The SIS process in some graphs may be not unimodal, implying that there is more than one nite time at which etremes (satisfying de[x i(t)] = 0 and (6)) are reached. 4 Bounds In addition to a general conveity bound (29) derived in Appendi B, we will deduce an upper and lower bound for Pr [X i (t) = ] by bounding () where i (; t) ea ik (u) du ea ki (u) Pr [X k (u) = jx i (u) = 0] du i (; t) ea ik (u) du i (; t) = i (; t) = ma Pr [X k (u) = jx i (u) = 0] (2) k2n i and u2[;t] min Pr [X k (u) = jx i (u) = 0] (3) k2n i and u2[;t] and N i denotes the set of neighbors of node i. Thus, i (; t) (and similarly for i (; t) ) is the probability of occurrence of the most (least) infectious link incident to a healthy node i during Non-unimodality was rst observed by Joel Miller and communicated to me. 5

6 the time interval [; t]. The de nition (2) for ed and small = c and large t = T suggests that the higher i (c; T ) is, the better node i is reachable. Hence, i (c; T ) can be considered as a centrality or importance measure for node i, such as betweenness, closeness and the diagonal element of the pseudo-inverse of the Laplacian [2] and suggests that i (c; T ) may be approimated by topology information from the adjacency matri A. By the non-negative SIS correlation property [24], we have Pr [X k (u) = jx i (u) = 0] Pr [X k (u) = ], so that i (; t) The minimum i (; t) in (7) produces the upper bound Pr [X i (t) = 0] i (s) e ma Pr [X k (u) = ] (4) k2n i and u2[;t] s f i (u)+ i (;t) e d i (u)gdu ds + Pr [X i () = 0] e f i (u)+ i (;t) e d i (u)gdu where d e i (u) = P N k= ea ik (u) is the time-dependent, weighted degree of node i. The maimum i (; t) in (7) produces the lower bound We use Pr [X i (t) = 0] i (s) e s f i (u)+ i (;t) e d i (u)gdu ds + Pr [X i () = 0] e i (s) er s f i (u)+ i (;t) e d i (u)gdu ds = e f i (u)+ i (;t) e d i (u)gdu i (; t) f i (u)+ i (;t) e d i (u)gdu ed i (s) er s 0 f i (u)+ i (;t) e d i (u)gdu ds in the above bounds for Pr [X i (t) = 0] to obtain bounds for the probability Pr [X i (t) = ] that node i is infected at time t: Pr [X i (t) = ] i (; t) Pr [X i (t) = ] i (; t) ed i (s) e ed i (s) e s f i (u)+ i (;t) e d i (u)gdu ds + Pr [X i () = ] e s f i (u)+ i (;t) e d i (u)gdu ds + Pr [X i () = ] e f i (u)+ i (;t) e d i (u)gdu f i (u)+ i (;t) e d i (u)gdu The rst integral over s in the above bounds is always positive, but generally di cult to evaluate. The inequalities for the homogenous case with constant rates simplify to Pr [X i (t) = ] i (; t) d i + e (+ i(;t)d i )(t ) Pr [X i () = ] + i (; t) d i Pr [X i (t) = ] i (; t) d i + e (+ i(;t)d i )(t ) Pr [X i () = ] + i (; t) d i i (; t) d i + i (; t) d i i (; t) d i + i (; t) d i with equality at t =. The upper bound (analogously for the lower bound) only eceeds (5) (6) i (;t)d i + i (;t)d i for nite time t provided the infection probability at time is Pr [X i () = ] > i(;t)d i + i (;t)d i. These bounds (5) and (6) resemble the time-dependent solution of a general continuous-time two-state Markov process [5, p. 23], which is also the underlying basis for NIMFA as eplained in [2, Sec. 2. and Fig..]. After su ciently large time t = T to ignore the initial condition at time (i.e. e T Pr [X i () = ] << ), the inequalities (5) and (6) simply with = to i (; T ) d i + i (; T ) d i Pr [X i (T ) = ] 6 i (; T ) d i + i (; T ) d i (7)

7 Similarly, the etremal probability (6) for constant rates can be bounded as i (; ) d i + i (; ) d i Pr [X i () = ] i (; ) d i + i (; ) d i The correspondence with (7) indicates that the quasi-stationary (or metastable) SIS regime attained after time T operates around the maimal value within bounds depending on infectious links. The smaller the di erence i (; T ) i (; T ), the sharper and more promising the above bounds will be. Also, (7) supports the earlier claim based upon () that the nodal infection probability can be approimated by Pr [X i (t) = ] r i i (t)+r i. De ning the maimum possible infection probability v ma = ma k2n and u2[0;t ] Pr [X k (u) = ] in the interval [0; T ], then the upper bound on i (; t) in (4) shows that i (0; T ) v ma and the upper bound in (7) is v mad i Pr [X i (T ) = ] + v ma d i whereas the corresponding upper bound for the steady-state infection probability v i in NIMFA [5, Theorem on p. 464] obeys v i d i + d i If v ma occurs at node i, the above upper bound becomes v ma d i, v ma d i d ma vmad i +v mad i and, equivalently for For any regular graph with degree r, we thus observe that Pr [X i (T ) = ] r, while equality holds in NIMFA. Despite the lack of a companion of (4) for the minimum min = min i2n i (0; t) in (3), we can proceed similarly, min d i + min d i Pr [X i (T ) = ] If the minimum possible infection probability v min in the interval [0; T ] occurs at node k, then min d min + min d min mind k + min d k v min from which an upper bound for the e ective infection rate follows as min d min v min v min (8) More equations for the e ective infection rate of this type are presented in Appendi C. 5 Summary Integral equations (7) and (9) for the most general Markovian setting of SIS epidemics are derived. The analysis illustrates that only the N rst, linear Markovian equations (, 4, 3) for the nodal infection probabilities alone out of the 2 N other equations for joint probabilities are elegantly bounded by the etremal probability form (6,) and the bounds resemble the behavior of the corresponding N non-linear NIMFA equations. 7

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9 [24] E. Cator and P. Van Mieghem. Nodal infection in Markovian SIS and SIR epidemics on networks are non-negatively correlated. Physical Review E, 89(5):052802, 204. [25] P. M. Morse and H. Feshbach. Methods of Theoretical Physics. McGraw-Hill Book Company, New York, 978. [26] P. Van Mieghem. SIS epidemics with time-dependent rates describing ageing of information spread and mutation of pathogens. Delft University of Technology, Report ( [27] P. Van Mieghem, F. D. Sahneh, and C. Scoglio. Eact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold. Proceedings of the 53rd IEEE Conference on Decision and Control (CDC204), December 5-7, Los Angeles, CA, USA, 204. [28] P. Van Mieghem. Approimate formula and bounds for the time-varying SIS prevalence in networks. Physical Review E, 93(5):05232, 206. [29] Q. Liu and P. Van Mieghem. Evaluation of an analytic, approimate formula for the time-varying SIS prevalence in di erent networks. Physica A, 47: , 207. [30] P. Van Mieghem and K. Devrien. An Epidemic Perspective on the Cut Size in Networks. Delft University of Technology, Report ( A Linear rst order di erential equation We rewrite (4) as df (t) + g (t) f (t) = g (t) i (t) (9) where f (t) = E [X i (t)] and g (t) = i (t) + b i (t) = G (f (t)), where G (:) is an unknown function. Since the dependence of g (t) on f (t) is unknown, the solution of the di erential equation (9) (see e.g. [25, Sec. 5.2]) f (t) = ep 0 Z s g (u) du ep g (u) du (g (s) i (s)) ds + f (0) 0 0 epresses f (t) in terms of g (t) and thus results in an integral equation for f (t), rather than in its solution. After integrating the rst term, Z s ep g (u) du g (s) ds = ep g (u) du and some rearrangements, we arrive at f (t) = (s) ep 0 s g (u) du ds + f f (0)g ep We rewrite the di erential equation (4) with the de nition b i (t) in (5) as 0 g (u) du d Pr [X i (t) = 0] + f i (t) + b i (t)g Pr [X i (t) = 0] = i (t) (22) Only if is a constant (independent of time) and assuming that G (h (t)) = i + b i (t) eists and is known, then the above di erential equation becomes with h (t) = Pr [X i (t) = 0] = dh (t) + G (h (t)) h (t) = i f (t), (20) (2) 9

10 whose solution is Z h(t) h(0) i d G () = t Let dh() d = i G(), then H (h (t)) H (h (0)) = t and the eplicit solution is h (t) = H (t + H (h (0))). For the particular eample where g (t) = f (t), the di erential equation (9) becomes a generalized logistic or non-linear Bernoulli di erential equation df(t) f (t) = f 2 (t), which describes, in the mean- eld approimation, the probability v (t) of infection in a node at time t in a regular graph G with degree r obeying dv (t) = r (t) v (t) ( v (t)) (t) v (t) (23) where the infection rate (t) and the curing rate (t) are general non-negative real functions of time t. By the same technique variation of a constant, we nd in [26] the solution R t ep 0 (r (u) (u)) du v (t) = v 0 + r 0 (s) ep R s 0 (r (u) (u)) du ds where v 0 is the initial fraction of infected nodes. (24) B The function p i (t; q) and proofs of (9) We give two proofs of (9): after introducing the function p i (t; q) in (25), the rst proof transforms the integral equation (7) into (9), while the second proof of (9) is based on the governing SIS equation (). Proof : Let us denote p i (t; q) = Pr [X i (t) = 0] e q f i(u)+b i (u)gdu (25) where q is an arbitrary point in time, then the integral equation (7) can be recast as p i (t; q) p i (; q) = i (s) er s q f i(u)+b i (u)gdu ds After invoking the de nition (25), we obtain an integral equation in p i (t; q), p i (t; q) p i (; q) = i (s) p i (s; q) ds (26) Pr [X i (s) = 0] Since the integrand is non-negative, then p i (t; q) p i (; q) for t > and p i (t; q) is non-decreasing in time t. Given p i (; q), we can iterate the integral equation (26). After n iterations, we obtain the series of multiple integrals, Z p i (t; q) t p i (; q) = i (t ) Pr [X i (t ) = 0] + i (t ) Pr [X i (t ) = 0] i (t ) Pr [X i (t ) = 0] n p i (t n ; q) p i (; q) i (t 2 ) 2 Pr [X i (t 2 ) = 0] i (t n ) n Pr [X i (t n ) = 0] 0

11 Using the multiple integral formula (which can be veri ed by partial integration ) yields i (t ) Pr [X i (t ) = 0] n p i (t; q) p i (; q) = + X k= Z t k! The last integral can be bounded as Z i (t ) tn Pr [X i (t ) = 0] Z i (t 2 ) tn 2 Pr [X i (t 2 ) = 0] k Z i (s) ds t + Pr [X i (s) = 0] p i (t n ; q) p i (; q) i (t n ) n Pr [X i (t n ) = 0] = n! Z i (t ) tn Pr [X i (t ) = 0] i (t n ) n Pr [X i (t n ) = 0] p Z i (t; q) t p i (; q) n! n i (s) ds Pr [X i (s) = 0] p i (t n ; q) p i (; q) and illustrates that the series converges in any nite interval when n!. Hence, p i (t; q) p i (; q) = + X k= Z t k! k Z i (s) ds t = ep Pr [X i (s) = 0] i (t n ) n Pr [X i (t n ) = 0] n i (s) ds Pr [X i (s) = 0] i (s) ds Pr [X i (s) = 0] Introducing the de nition (25) of p i (t; ) results in which can be rewritten as (9). Pr [X i (t) = 0] e f i(u)+b i (u)gdu Pr [X i () = 0] i (s) ds = ep Pr [X i (s) = 0] Proof 2: We can generalize (9) to any integrable function h () = dh() d as (27) H (Pr [X i (t) = 0]) H (Pr [X i () = 0]) = h (Pr [X i (u) = 0]) d Pr [X i (u) = 0] du du Introducing the governing SIS equation (22) for the healthy probability Pr [X i (u) = 0] leads to the general formula H (q)j Pr[X i(t)=0] Pr[X i ()=0] = h (Pr [X i (u) = 0]) i (u) Pr [X i (u) = ] where the special case h () = leads to (9).! ea ki (u) Pr [X k (u) = ; X i (u) = 0] du k= (28) Lemma Provided that the curing rate i (t), the function p t+ i (t; q), de ned in (25), is i (0) conve for all time t. Proof: Di erentiating (26) with respect to time t yields dp i (t; q) = i (t) Pr [X i (t) = 0] p i (t; q) A second di erentiation, in which we use the above rst order di erential, leads to ( d 2 ) p i (t; q) d i (t) i (t) 2 2 = + p i (t; q) Pr [X i (t) = 0] Pr [X i (t) = 0]

12 and d 2 p i (t; q) 2 = i (t) + 0 i (t) i (t) Pr [X i (t) = 0] Introducing the governing equation (22) yields d Pr [X i (t) = 0] i (t) p i (t; q) (Pr [X i (t) = 0]) 2 d 2 p i (t; q) 2 = 0 i (t) i (t) + i (t) + b i (t) i (t) p i (t; q) Pr [X i (t) = 0] which is always non-negative provided 0 i (t) i (t) + i (t) 0, which is, after integration, equivalent to the condition stated in Lemma. As a consequence of Lemma, we conclude for a constant curing rate i (t) = i that p i (t; q) is always conve. Applying the conveity de nition (see e.g. [5, p. 0]), p i (u + ( ) v; q) p i (u; q) + ( ) p i (v; q) with 2 [0; ] where i (t) obeys the condition in Lemma, then the epression (27), after choosing = u, leads to a bound involving the healthy probability where the time e = u + ( ) v 2 [u; v]: e R u+( u )v i (s)ds Pr[X i (s)=0] + ( ) er v u i (s)ds Pr[X i (s)=0] (29) C The prevalence and e ective infection rate Summing (3) over all nodes and rewritten in terms of the prevalence " # y (t) = Pr [X i (t) = ] = E X i (t) N N i= de ned as the epected fraction of infected nodes in a graph at time t, yields i= (30) dy (t) + N i (t) E [X i (t)] = N i= ea ki (t) Pr [X k = ; X i = 0] i= k= Only when all curing rates i (t) = (t) are equal, we nd a di erential equation in the prevalence dy (t) + (t) y (t) = N ea ki (t) Pr [X k = ; X i = 0] i= k= Earlier, we have demonstrated in [27] and further studied in [28, 29], that dy (t ; ) = y (t ; ) + h i N E w (t ; ) T Qw (t ; ) where t = t is the scaled time, Q = A is the Laplacian of the graph G with = diag(d ; d 2 ; : : : ; d N ) and d i is the degree of node i in G, and the Bernoulli vector w = (X ; X 2 ; : : : ; X N ). If we con ne ourselves for simplicity to ed rates and a time-invariant topology, then the comparison with (3) shows that the average number of links in the cut-set between healthy and infected nodes is (3) E w T Qw = a ki Pr [X k = ; X i = 0] (32) i= k= 2

13 which is investigated further in [30]. In the metastable regime, the homogeneous case with ed rates ( ij (t) =, i (t) = ) and time-invariant topology a ki (t) = a ki in (6) with (5) satis es to a very good approimation Pr [X i () = ] = P N k= a ki Pr [X k () = jx i () = 0] + P N k= a ki Pr [X k () = jx i () = 0] from which the e ective infection rate = is solved as = Pr [X i () = ] P N k= a ki Pr [X k () = ; X i () = 0] (33) The e ective infection rate can also be solved 2 from (3) with (32) in terms of the prevalence as = y () N P N i= P N k= a ki Pr [X k () = ; X i () = 0] (34) We revisit the deduction 3 from [27] and rewrite (34) as = P N k= Pr [X k () = ] P N i= a ki Pr [X i () = 0jX k () = ] P N k= Pr [X k () = ] for a time, somewhere in the metastable regime (see Section 3.). Now Pr [X i (t) = 0jX k (t) = ] = Pr [X i (t) = 0] Pr [X k (t) = ] Pr [X k (t) = jx i (t) = 0] appears as the probability that node i is healthy given an infected neighbor k, which opposes the dual conditional probability Pr [X k (t) = jx i (t) = 0] above and both conditional probabilities are only equal when Pr [X i (t) = 0] = Pr [X k (t) = ], i.e. when both nodes have equal probability to be in an opposite state at any time t. Upper and lower bounding in the usual way leads to min kn i= a ki Pr [X i () = 0jX k () = ] ma kn Using the degree d i = P N j= a ij and con ning to the lower bound, we have min kn i= a ki Pr [X i () = 0jX k () = ] a ki Pr [X i () = 0jX k () = ] min min Pr [X i () = 0jX l () = ] d k kn (i;l)2l After a similar treatment of the upper bound, we arrive at i= d min min Pr [X i () = 0jX l () = ] d ma ma Pr [X i () = 0jX k () = ] (35) (i;l)2l (i;l)2l We remark that min = min i2n i () = min (i;l)2l and u2[0;] Pr [X k (u) = jx i (u) = 0], epressed in dual conditional probability, appears in (8) and is di erent from the above. 2 Also by summing (33) over all nodes after obvious manipulations. 3 Earlier, (34) has already appeared in the proof of Theorem in [5, p. 458] and in [27]. The last part of the proof in [5, p. 459] was, unfortunately erroneous and thus also Theorem 7.3.2, as reported in [28]. 3