Abstract— In this paper we study the global dynamics of epidemic spread over complex networks for both discrete-time and continuous-time models. In this setting, the state of the system at any given time is the vector obtained from the marginal probability of infection of each of the nodes in the network at that time. Convergence to the origin means that the epidemic eventually dies out. Linearizing the model around the origin yields a system whose state is an upper bound on the true state. As a result, whenever the linearized model is locally stable, the original model is globally stable, with the origin being its fixed point. When the linearized model is unstable the origin is not a stable fixed point and we show the existence of a unique second fixed point. In the continuous-time model, this second fixed point attracts all points in the state space other than the origin. In the discrete-time setting we consider two models. In the first model, we show that the second fixed point attracts all points in the state space other than the origin. In the second model, however, we show this need not be the case. We therefore give conditions under which the second fixed point attracts all non-origin points and show that for random Erd¨os-R´enyi graphs this happens with high probability.

I. INTRODUCTION Modeling the spread of an infectious disease on a complex network is a topic of increasing interest. The classical epidemic models include the susceptible-infected-susceptible (SIS) model. In the SIS model, each node of the network exists in one of two different states. In the susceptible state, the node is healthy but it may be infected if it is exposed to disease. The node is exposed to disease if any of its neighbors in the network are infected. The probability for being infected from a susceptible state depends on the number of infected neighbors. From an infected state, the node may become healthy with some probability, after which it returns to a susceptible state. Early work in the SIS model has been conducted using random graph approach [12], [13]. In the random graph model, the number of infected neighbors depend on both of the degree distribution and rate of infected nodes. The work is focused on high-probability behavior of steady state and convergence of various random graph models. The other approach is fixed graph one. We can model this process as Markov chain with 2n states where n is the number of nodes in the network when underlying graph is given [4], [7], [11]. At each time, the state of the Markov chain represents the joint probability of the all the nodes being in either susceptible or infected states. Since the joint

probability distribution at any given time only depends on the joint probability distribution at the previous time this is a Markov chain. The transition matrix of the Markov chain is closely related to the adjacency matrix of the underlying graph. When the underlying graph is connected, PerronFrobenius theory implies that the Markov chain has a unique absorbing state, which is the state where all the nodes in the network are susceptible. The reason being that if all the nodes are susceptible, no node will be exposed to disease, and therefore they will always stay healthy. It is therefore clear that, in the Markov chain model, if one waits long enough, the epidemic will die out and all nodes will be healthy. However, this result is not very informative, since depending on the mixing time of the Markov chain it may take a very long time, in fact exponentially long in the number of nodes, for steady state to be achieved. In this situation, the dying out of the epidemic may not be observed in practice. As a result of this, we study approximate models which have a much smaller number of states. Once such model has been introduced by Wang et. al. and Chakrabarti et. al. [17], [3] and some more work has been conducted based on the model [5], [8]. In what follows we shall study this and two other models (a discrete-time model similar to the Chakrabarti model, as well as a continuous-time model). The main result is that, when the origin (corresponding to the all susceptible states) is not stable, the epidemic model has a unique second fixed point. For the continuous-time model, and Chakrabarti model, we show that the second fixed point attracts all non-origin points. For the other discrete-time model, we show that this is not necessarily the case, and give conditions under which the second fixed point is stable. II. M ODEL D ESCRIPTION For a given connected network G with n nodes, let Ni be the neighborhood of node i. Let A be the adjacency matrix of G. Pi (t) is the probability that node i is infected at time t. Let δ be probability of recovery when a node is infected, and β be the probability to be infected from one of one’s infected neighbors. Assuming the event of being infected from a neighboring node happens independently for each neighboring node, we may write !

1 Department

of Computing and Mathematical Sciences, California institute of Technology, 1200 E California Blvd, CA91125, USA ctznahj

Pi (t + 1) = Pi (t)(1 − δ ) + (1 − Pi (t)) 1 −

at caltech.edu

(1) This is the first discrete-time model for epidemic spread that we shall consider. To understand the behavior of this model,

2 Department

of Electrical Engineering, California institute of Technology, 1200 E California Blvd, CA91125, USA hassibi at

caltech.edu

∏ (1 − β Pj (t))

j∈Ni

we obtain the following upper bound of (1): ! Pi (t + 1) ≤ (1 − δ )Pi (t) + β (1 − Pi (t))

∑ Pj (t)

(2)

j∈Ni

! ≤ (1 − δ )Pi (t) + β

∑ Pj (t)

(3)

j∈Ni

(2) can be regarded as an approximation of (1) for small β . (3) is a linearized version of (1), where we have linearized around the origin. Note that this linearization gives an upper bound on the original model. For two real-valued column vectors u = (u1 , · · · , un )T , v = (v1 , · · · , vn )T ∈ Rn , we say u v if ui ≤ vi for all i ∈ {1, · · · , n} and u ≺ v if ui < vi for all i ∈ {1, · · · , n}. For P(t) = (P1 (t), · · · , Pn (t))T P(t + 1) ((1 − δ )In + β A)P(t)

(4)

It is obvious that Pi (t) converges to the origin for both of (1) and (2) if λmax ((1−δ )In +β A) < 1. The reason is that this happens for the linearized upper bound. We will therefore focus on the dynamics of the system when λmax ((1 − δ )In + β A) > 1. Wang et. al. [17] and Chakrabarti et. al. [3] focus on staying healthy by defining the probability that a node receives no infection from its neighborhood. We focus on infection rather than staying healthy. Let Ξ : [0, 1]n → [0, 1)n with Ξ = (Ξ1 , · · · , Ξn )T be a map associated with network G satisfying the three properties below. ∂ Ξi = β Ai, j at the origin. (a) Ξi (x) = 0 and ∂xj ∂ Ξi ∂ Ξi (b) > 0 if i ∈ N j in G, and = 0 if i ∈ / N j in G. ∂xj ∂xj ∂ 2 Ξi (c) For any i, j, k ∈ {1, · · · , n}, ≤ 0. ∂ x j ∂ xk It is obvious that Ξi (x) = 1 − ∏ j∈Ni (1 − β x j ) of (1) satisfies the conditions above. Ξi (x) = β ∑ j∈Ni x j of (2) also satisfies the condition if β is small enough. Ξi is the probability that node i will be infected from its neighbors in the next time-step. Let ω : [0, 1] → R+ be a function which also satisfies three properties below. (d) ω(0) = 0, ω(1) ≥ 1 (e) ω 0 (0) = δ , ω 0 (s) > 0 for all s ∈ (0, 1) ω(s1 ) ω(s2 ) < if s1 < s2 (f) s1 s2 δs It is also obvious that ω(s) = satisfies all three 1−s conditions above.

Lemma 2.1: Let hi,u,v : s → Ξi (u + sv) be a function defined on subset of nonnegative real numbers for given hi,u,v (s) − hi,u,v (0) i ∈ {1, · · · , n}, u, v ∈ [0, 1]n . Then is a s decreasing function. Proof: hi,u,v (s) is concave by property (c). d hi,u,v (s) − hi,u,v (0) ds s hi,u,v (s) − hi,u,v (0) 1 0 = hi,u,v (s) − (6) s s 1 0 = hi,u,v (s) − h0i,u,v (s∗ ) ≤ 0 (7) s hi,u,v (s) − hi,u,v (0) = h0i,u,v (s∗ ) for some s∗ ∈ (0, s) by the s mean value theorem. III. E XISTENCE AND U NIQUENESS OF N ONTRIVIAL F IXED P OINT The origin, the trivial fixed point of the system equation is unstable if λmax ((1−δ )In +β A) > 1. However, it is unknown whether the system has another fixed point or not. In this section, we prove that there actually exists a nontrivial fixed point of (5). We also prove that the nontrivial fixed point is unique. Lemma 3.1: λmax ((1 − δ )In + β A) > 1 if and only if there exists v (0, · · · , 0)T = 0n such that (β A − δ In )v 0n Proof: Suppose that λmax ((1 − δ )In + β A) > 1 and w as an eigenvector corresponding to the maximum eigenvalue. (1 − δ )In + β A is nonnegative and irreducible (a nonnegative matrix X is irreducible if there exists m(i, j) ∈ N for each m(i, j) pair of indices i, j such that Xi, j is nonzero.) because A is the adjacency matrix of a connected graph G. Every entry of w is positive by Perron-Frobenius theorem for irreducible matrices, and (β A − δ In )w 0n because the eigenvalue corresponding to w is greater than unity. Suppose that there exits v 0n such that (β A−δ In )v 0n . Then, ((1 − δ )In + β A)v v k((1 − δ )In + β A)uk2 kuk2 u∈Rn k((1 − δ )In + β A)vk2 ≥ >1 kvk2

λmax ((1 − δ )In + β A) = sup

(5)

We close this section by giving a lemma which is useful in the next section.

(9)

The main theorem of this section follows. Theorem 3.2: Define a map Ψ : [0, 1]n → Rn with Ξ and ω satisfying the conditions (a)-(f) above. Define Ψi (x) = Ξi (x) − ω(xi ) .

(10)

Then Ψ = (Ψ1 , · · · , Ψn ) has a unique nontrivial (other than the origin) zero if λmax ((1 − δ )In + β A) > 1. Proof: Si and S are defined by Ψ as below.

We can view (1) and (2) as Pi (t + 1) = Pi (t) + (1 − Pi (t))(Ξi (P(t)) − ω(Pi (t)))

(8)

Si = {x ∈ [0, 1]n : Ψi (x) ≥ 0}

S=

n \

Si

(11)

i=1

By the lemma above, there exists v 0n such that (β A − δ In )v 0n . There is a small ε > 0 such that εv ∈ S because

the Jacobian of Ψ = (Ψ1 , · · · , Ψn )T is equal to β A − δ In at the origin and Ψ(0) = 0 by property (a) of Ξ and (d) of ω. Define max(x, y) = (max(x1 , y1 ), · · · , max(xn , yn )). We claim that max(x, y) ∈ S if x, y ∈ S. The proof follows. max(xi , yi ) = xi without loss of generality for x, y ∈ S. Ψi (max(x, y)) = Ξi (max(x, y)) − ω(xi ) ≥ Ξi (x) − ω(xi ) ≥ 0 (12) The first inequality holds by property (b), and the second inequality holds because x ∈ S. Therefore max(x, y) ∈ Si for every i and it completes the proof of the claim. This leads to the existence of a unique maximal point x∗ ∈ S such that x∗ x for all x ∈ S. εv ∈ S and the maximality of x∗ guarantees that x∗ has positive entries. We claim that Ψi (x∗ ) = 0 for all i ∈ {1, · · · , n}. Assume that Ψi (x∗ ) 6= 0 for some i. Then, Ψi (x∗ ) > 0 since x∗ ∈ S. There exists zi > xi∗ such that Ψi (x∗ ) = Ξi (x∗ ) − ω(xi∗ ) > Ξi (x∗ ) − ω(zi ) ≥ 0 Define z = (z1 , · · · , zn k ∈ {1, · · · , n},

)T

with z j =

x∗j

(13)

for j 6= i. For every

Ψk (z) = Ξk (z) − ω(zk ) ≥ Ξk (x∗ ) − ω(zk ) ≥ 0

(14)

The first inequality of (14) holds by property (b). The second inequality of (14) holds by (13) if k = i and the inequality holds by definition of z if k 6= i. (14) guarantees that z ∈ S. zi > xi∗ and z j = x∗j for j 6= i contradict that x∗ is the maximal point of S. The assumption was therefore wrong, Ψi (x∗ ) = 0 for all i ∈ {1, · · · , n}, and there exists a nontrivial zero of Ψ. The next step is showing that x∗ is the unique nontrivial zero of Ψ. Assume that y∗ is another nontrivial zero. Then y∗ ∈ S and Ψ(y∗ ) = 0n . We claim that every entry of y∗ is positive. Define K0 and K+ where y∗i = 0 if i ∈ K0 and y∗i > 0 if i ∈ K+ . Then, K0 ∪ K+ = {1, · · · , n}. K0 and K+ are separation of vertex set of the system. Assume that K0 is a non-empty set. There exists j ∈ K+ such that j is connected to a node in K0 because G is connected. Denote k ∈ K0 as a node which is connected to j. Ψk (y∗ ) = Ξk (y∗ ) − ω(y∗k ) = Ξk (y∗ ) > 0 (15) The inequality above is strict by property (b) since k ∈ N j and y∗j > 0. It contradicts that Ψ(y∗ ) = 0. K0 is the empty set. We get the following inequality by Lemma 2.1 for u = 0n , v = x∗ and s ≤ 1. Ξi (sx∗ ) hi,u,v (s) − hi,u,v (0) = ≥ hi,u,v (1) − hi,u,v (0) = Ξi (x∗ ) s s (16) There exists α ∈ (0, 1) such that y∗ αx∗ and y∗j = αx∗j for some j ∈ {1, · · · , n}. Ψ j (y∗ ) = Ξ j (y∗ ) − ω(αx∗j ) ∗

≥ Ξ j (αx ) − ω(αx∗j ) ≥ αΞ j (x∗ ) − ω(αx∗j ) > α Ξ j (x∗ ) − ω(x∗j ) = 0

(17) (18) (19) (20)

(18) and (19) are guaranteed by property (b) and (16). (20) αω(x∗j ) ω(αx∗j ) also holds because > by α ∈ (0, 1), x∗j > 0 αx∗j αx∗j and property (f). This contradicts that Ψi (y∗ ) = 0 for all i. Therefore x∗ is the unique nontrivial zero of Ψ. We remark that x∗ in the above proof is also the unique nontrivial fixed point of (5). As a further remark, consider a network whose edge {i, j} has weight wi j = w ji ∈ [0, 1]. The weight of each edge could represent the degree of intimacy. The weight matrix can replace the adjacency matrix to define Ξi (x) = 1 − ∏ j∈Ni (1 − β wi j x j ) . Then Ξ defined by the weight matrix rather than the adjacency matrix also satisfies all three conditions (a)-(c) if Ai j is replaced by wi j from (a). The system of equations will still have the same properties even if we admit different weights. IV. L OCAL S TABILITY OF THE N ONTRIVIAL F IXED P OINT Let Φ : [0, 1]n → [0, 1]n be a map whose i-th component is defined as Φi (x) = (1 − δ )xi + (1 − xi )Ξi (x)

(21)

Φi (x) = xi + (1 − xi )Ψi (x) and P(t + 1) = Φ(P(t)). The origin, the trivial fixed point of the system is globally stable if λmax ((1 − δ )In + β A) < 1. The next issue is whether the unique nontrivial fixed point is also stable if λmax ((1 − δ )In + β A) > 1. This is not true in general. The following is an example of an unstable nontrivial fixed point. 0 1 1 A= 1 0 0 δ = 0.9 β = 0.9 (22) 1 0 0 The nontrivial fixed point of the system above is x∗ = (0.286, 0.222, 0.222)T . The Jacobian matrix of Φ at x∗ is −0.260 0.514 0.514 0 (23) JΦ(x∗ ) = 0.700 −0.157 0.700 0 −0.157 The eigenvalue with largest absolute value in the above Jacobian matrix is −1.059 whose absolute value is greater than 1. However, P(t) converges to a cycle rather than a nontrivial fixed point x∗ . A. Random Graphs Even though the nontrivial fixed point is not stable generally, we shall show that it is stable with high probability for random Erd¨os-R´enyi graphs. To study the stability of the nontrivial fixed point with high probability, we will begin with the following lemma that demonstrates that the Jacobian matrix at x∗ has no eigenvalue greater than or equal to unity for any values of β and δ and for any connected graph. Lemma 4.1: Suppose that x∗ is a unique nontrivial fixed point of Φ : [0, 1]n → [0, 1]n with Ξ satisfying the conditions (a),(b) and (c) when λmax ((1 − δ )In + β A) > 1. Then the

Jacobian matrix of Φ at x∗ has no eigenvalue which is greater than or equal to 1. Proof: JΦ = (1 − δ )In − diag(Ξ) + diag(1n − x)JΞ

(24)

JΦ + δ In is a nonnegative matrix by some properties of Ξ. By the Perron-Frobenius theorem of irreducible aperiodic matrices, JΦ + δ In has an eigenvector v with eigenvalue µ whose components are all positive, and any other eigenvalues of JΦ + δ In is strictly smaller than µ in absolute value. For each eigenvalue λ of JΦ , there is λ 0 which is an eigenvalue of JΦ + δ In and λ 0 = λ + δ . Furthermore, the eigenvectors corresponding to λ and λ 0 are the same. JΦ has the largest eigenvalue κ = µ − δ corresponding to eigenvector v. All the entries of v are strictly positive. δs in this section, and define a map Ω : Fix ω(s) = 1−s [0, 1]n → Rn+ by Ω(x1 , · · · , xn ) = (ω(x1 ), · · · , ω(xn ))T . δ + Ξi (x∗ ) = δ +

δ xi∗ δ = = (1 − xi∗ )ω 0 (xi∗ ) 1 − xi∗ 1 − xi∗

(25)

By applying (25) to (24), JΦ(x∗ ) = In + diag(1n − x∗ ) JΞ(x∗ ) − JΩ(x∗ )

(26)

Assume that κ > 1. κv = JΦ(x∗ ) v = v + diag(1n − x∗ ) JΞ(x∗ ) − JΩ(x∗ ) v (27) JΨ(x∗ ) v = JΞ(x∗ ) − JΩ(x∗ ) v = (κ −1) ( diag(1n − x∗ ))−1 v 0n (28) Ψ(x∗ + v) ≈ Ψ(x∗ ) + JΨ(x∗ ) v 0n (29) There is an ε > 0 which is small enough and Ψ(x∗ +εv) 0n . However, it contradicts that x∗ is the maximal element in S. Therefore the assumption was wrong. Assume now that κ = 1. JΨ(x∗ ) v = JΞ(x∗ ) − JΩ(x∗ ) v = (κ −1) ( diag(1n − x∗ ))−1 v = 0n (30) δ 00 By property (c) of Ξ and ω (s) = > 0, the Hes(1 − s)3 sian matrix of Ψi is a non-positive matrix. Furthermore, δ (HΨi (x∗ ) )i,i = −ω 00 (xi∗ ) = − < 0, (1 − xi∗ )3 1 Ψi (x∗ + v) ≈ Ψi (x∗ ) + grad(Ψi (x∗ ))v + vT HΨi (x∗ ) v < 0 2 (31) Since the inequality above holds for every i ∈ {1, · · · , n}, there exists ε > 0 which is small and such that the entries of Ψ(x∗ − εv) are all negative. There is α ∈ (0, 1) such that x∗ − εv αx∗ and (x∗ − εv) j = αx∗j for some j ∈ {1, · · · , n}. Just substitute x∗ − εv for y∗ in (19). Then, Ψ j (x∗ − εv) is positive, however it contradicts that Ψ(x∗ − εv) has all negative entries. Therefore the assumption was wrong. Even though JΦ has no eigenvalue which is greater than or equal to 1, the fixed point x∗ still has a chance to be unstable if there is an eigenvalue which is greater than or equal to 1 in absolute value. We now show that x∗ is stable with high probability when we consider a certain family of

random graphs and the number of vertices is large. We will later show that this family of random graphs includes Erd¨osR´enyi graphs. We apply Ξi (x) = 1 − ∏ j∈Ni (1 − β x j ) of (1) from now on. ∂ Ξi 1 − Ξi = β ∏ (1 − β xk ) = β if i ∈ N j in G (32) ∂xj 1 −βxj k∈N \{ j} i

JΞ = β diag(1n − Ξ)A diag(1n − β x)−1

(33)

Lemma 4.2: Suppose that G(n) is a random graph with (n) (n) n vertices and let dmin and dmax denote the minimum and (n) (n) maximum degree of G(n) . If Pr[(dmin )2 > a · dmax ] goes to 1 as n goes to infinity for any fixed a > 0, then the system is unstable at the origin and locally stable at the nontrivial fixed point x∗ with high probability as n grows, for any fixed β and δ . Proof: First note that λmax ((1 − δ )In + β A) ≥ (1 − δ ) + (n) (n) (n) β dmin . Since Pr[(dmin )2 > a · dmax ] goes to 1 as n goes to (n) infinity, this means Pr[dmin > a] goes to 1 as n goes to infinity, for any a, which further means that λmax exceeds one with high probability. Thus, the origin is unstable. JΦ = (1 − δ )In − diag(Ξ) + diag(1n − x)JΞ 1 2

' (1 − δ )In − diag(Ξ) + β D AD

1 2

(34) (35)

where D is a diagonal matrix whose i−th diagonal element (1 − xi )(1 − Ξi ) is and ' refers to similarity. 1 − β xi All the eigenvalues of JΦ are real because it is similar to a symmetric matrix by (35). Since Lemma 4.1 shows that all the eigenvalue of JΦ(x∗ ) are strictly less than 1, we need to show that all the eigenvalues of JΦ(x∗ ) + In are positive to show the system is locally stable at x∗ . By applying (25) to (35), 1

1

JΦ(x∗ ) + In ' 2In − δ diag(1n − x∗ )−1 + β D 2 AD 2

(36)

Since the right hand side of (36) is symmetric, it is positive definite if all the eigenvalues of JΦ(x∗ ) +In are strictly positive. With simple algebra, we can show that the right hand side of (36) is positive definite if and only if E + A is positive 1 1 1 definite where E = D− 2 (2In − δ diag(1n − x∗ )−1 )D− 2 is β a diagonal matrix whose i−th diagonal entry is defined as below. δ ∗ ∗ 2 1 − 2 − xi (1 − β xi ) Eii = · (37) β (1 − (1 + δ )xi∗ )(1 − xi∗ ) 2(1 − β xi∗ ) δ (1 − δ ) = · 1+ (38) β (1 + δ )(1 − xi∗ ) 2(1 − (1 + δ )xi∗ ) We will give a lower bound on Eii because the smallest eigenvalue of E + A is what we are interested in. 1 − (1 + δ )xi∗ δ xi∗ = 1− = 1 − Ξi (x∗ ) = ∗ 1 − xi 1 − xi∗

∏ (1 − β x∗j )

j∈Ni

(39)

∗ = min{x∗ , · · · , x∗ } and denote by d the degree of Let xm m n 1 the m-th node. ∗ ∗ 1 − (1 + δ )xm = (1 − xm )

∏ (1 − β x∗j ) ≤ (1 − β xm∗ )dm

(40)

j∈Nm

∗ > y. If y > 0 satisfies 1 − (1 + δ )y = (1 − β y)dm , then xm For 1 − (1 + δ )y = η, dm β (1 − η) (41) η = 1− 1+δ 1 β ⇔ 1 − η dm = (1 − η) (42) 1+δ 1 1 + δ dm −1 dk (43) ⇔ = ∑ η m ≥ 1 + (dm − 1)η 2 β k=0 1−β +δ 2 1−β +δ 2 ⇒ η≤ ≤ (44) β (dm − 1) β (dmin − 1)

dm and dmin are generally different. dm is the degree of the m-th node and dmin is a minimum degree of the network. δ (1 − δ ) 1 2 (45) · 1+ · Eii ≥ β (1 + δ ) 2 1 − (1 + δ )xi∗ 2 δ (1 − δ ) 1 ≥ · 1+ · (46) β (1 + δ ) 2 η ! 2 β (dmin − 1) 2 δ (1 − δ ) ≥ (47) · 1+ · β (1 + δ ) 2 1−β +δ Since λmin (A) ≥ −λmax (A) ≥ −dmax , 2 δ (1 − δ ) β (dmin − 1) 2 λmin (E +A) ≥ + · −dmax β (1 + δ ) β (1 + δ ) 1−β +δ (48) (48) guarantees that the smallest eigenvalue of E + A is positive and the system is locally stable at the nontrivial 2 fixed point with high probability if dmin grows faster than dmax as the size of graph grows. We can think of several random graph models that satisfy the condition of Lemma 4.2. For example, if the random graph has uniform degree then the minimum degree and maximum degree are identical and as long as the degree d2 grows with n, the ratio min = d will grow with any n dmax and exceed a with high probability. Similarly,for random graphs where the degree distribution of each node is identical and the degree distribution ”concentrates”, so that we can expect that the maximum degree and the minimum degree are proportional to the expected of degree, in which case 2 dmin grows if the expected degree increases unbounded with d n.max 1) Erd¨os-R´enyi Graphs: The Erd¨os-R´enyi random graph, G(n) = G(n, p(n)) has identical degree distribution. Theorem 4.3: Consider an Erd¨os-R´enyi random graph log n (n) G = G(n, p(n)) with p(n) = c where c > 1 is a n constant. Then (1) is locally unstable at the origin and has a locally stable nontrivial fixed point with high probability for any fixed β and δ .

Proof: The proof of instability of the origin is similar to the proof of Lemma 4.2. For the remaining, it is enough to (n) (n) show that Pr[(dmin )2 > a · dmax ] goes to 1 as n goes to infinity for any a > 0. The degree of random Erd¨os-R´enyi graphs is studied in [2]. In particular, there exists two constants η∆ > 0 and ηδ ∈ (−1, 0) such that (n)

dmax ∼ (1 + η∆ )c log n

(n)

dmin ∼ (1 + ηδ )c log n (n)

(49)

(n)

It is straightforward to see that Pr[(dmin )2 > a · dmax ] goes to 1 as n goes to infinity for any a > 0. log n for c = 1 is also the threshold for Since p = c n connectivity, we can say that connected Erd¨os-R´enyi graphs have a nontrivial stable fixed point with high probability. The Random geometric graph G(n) = G(n, r(n)) also has identical degree distribution if each node is distributed uniformly. As studied in [15], random such graphs have maximum and minimum degree which are proportional to the expected degree with high probability if r(n) is smaller than the threshold of connectivity. Like Erd¨os-R´enyi graphs, it has high probability of having a nontrivial stable fixed point if the degree grows with n. On the other hand, the minimum degree of the Barab´asiAlbert model is fixed as the size of the graph increases. In this case, we cannot generally argue that the nontrivial fixed point is stable. V. G LOBAL S TABILITY OF C HAKRABARTI M ODEL In this section we analyze discrete-time epidemic equation which is suggested by Chakrabarti, [3]. ! 1 − Pi (t + 1) = (δ Pi (t) + 1 − Pi (t))

∏ (1 − β Pj (t))

(50)

j∈Ni

The difference with (1) is that in (50) an infected node still has a chance of being infected from its neighborhood even though it recovers from the disease in the same time interval. In (1) an infected node is susceptible for at least one time interval after it recovers from the disease. With Ξi (x) = 1 − ∏ (1 − β x j ), the following equation is j∈Ni

equivalent to (50). Pi (t + 1) = (1 − δ )Pi (t) + (1 − (1 − δ )Pi (t))Ξi (P(t)) (51) δs satisfy all the properties from (a) 1 − (1 − δ )s to (f). The equation also has a unique nontrivial fixed point. e : [0, 1]n → [0, 1]n be a map whose i-th component is Let Φ defined as

Ξ and ω(s) =

e i (x) = (1 − δ )xi + (1 − (1 − δ )xi )Ξi (x) Φ

(52)

e P(t + 1) = Φ(P(t)) in the Chakrabarti model. Theorem 5.1: Suppose that λmax ((1−δ )In +β A) > 1, then P(t) defined by (51) converges to x∗ which is a nontrivial fixed point of (52) as t increases if P(0) is not the origin. ei ∂Φ Proof: It is trivial to check that ≥ 0 for any i, j ∈ ∂xj {1, · · · , n}.

e e Φ(x)) e e e is Suppose that Φ(x) x. Then, Φ( Φ(x) since Φ e e e e increasing. Similarly, Φ(Φ(x)) Φ(x) if Φ(x) x. Define a sequence y(0) = 1n = (1, 1, · · · , 1)T and y(k+1) = e Φ(y(k) ). y(1) = (1 − δ )1n + δ Ξ(1n ) 1n = y(0)

(53)

The equation above implies that y(k+1) y(k) for every k ∈ N. n The sequence {y(k) }∞ k=0 ⊂ [0, 1] has a limit point because it is decreasing, and bounded from below. Denote y∗ as a e ∗ ) = y∗ . There are two limit point of the sequence, then Φ(y ∗ e candidates for y because Φ has only two fixed points. e is an increasing map, and y(0) x for every x ∈ Since Φ n (k) e k (x). y(k) Φ e k (x∗ ) = x∗ for every k implies [0, 1] , y Φ ∗ ∗ that y x . It also implies that y∗ = x∗ . For any P(0) ∈ [0, 1]n , an upper bound of P(t) is y(t) and it goes to x∗ as t goes to infinity. Suppose that all the entries of P(0) are positive. This is reasonable since there exists m such that all the entries of P(m) are positive if P(0) is not the origin. There exists α ∈ (0, 1) such that αx∗ P(0). Define a sequence z(0) = αx∗ e (k) ). and z(k+1) = Φ(z δ αxi∗ (0) (0) (1) ∗ zi = zi + (1 − (1 − δ )zi ) Ξi (αx ) − 1 − (1 − δ )αxi∗ (54) ∗ δ xi (0) (0) > zi + α(1 − (1 − δ )zi ) Ξi (x∗ ) − 1 − (1 − δ )xi∗ (55) (0)

= zi

(56)

(57). The origin is unstable if, and only if, −δ In + β A has an eigenvalue in the RHP. Since A is symmetric and by PerronFrobenius theorem its largest eigenvalue in absolute value is positive, this is equivalent to λmax ((1 − δ )In + β A) > 1. By the previous arguments we know that (57) has a nontrivial fixed point x∗ under this condition. Theorem 6.1: Suppose that λmax ((1−δ )In +β A) > 1, then x(t) defined by (57) converges to x∗ as t goes to infinity unless x(0) = 0n Proof: We will suggest a Lyapunov function that is strictly decreasing for all initial points except x(0) = 0n : |xi − xi∗ | (58) V (x) = max 1≤i≤n xi∗ It’s obvious that V (x∗ ) = 0 and V (x) > 0 for all x ∈ [0, 1]n \ {x∗ }. Suppose that V (x) = r > 0. Then, x j ∈ [(1 − r)x∗j , (1 + r)x∗j ] for all j ∈ {1, · · · , n}. There is i such that xi = (1 − r)xi∗ or (1 + r)xi∗ . In the case of xi = (1 + r)xi∗ , 0n max(x, x∗ ) − x∗ rx∗ , we obtain the following equation by Lemma 2.1 for u = max(x, x∗ ) − x∗ max(x, x∗ ) − x∗ ,v = . x∗ − r r Ξi (x) ≤ Ξi (max(x, x∗ )) = Ξi (u + (1 + r)v) (59) hi,u,v (1 + r) − hi,u,v (0) + hi,u,v (0) (60) = (1 + r) 1+r ≤ (1 + r) (hi,u,v (1) − hi,u,v (0)) + hi,u,v (0) (61) = (1 + r)Ξi (u + v) − rΞi (u)

(62) ∗

≤ (1 + r)Ξi (u + v) = (1 + r)Ξi (x )

(63)

The equation above is necessary to prove following inequality: 1 − xi dxi = (Ξi (x) − ω(xi )) (64) dt ∆t 1 − (1 + r)xi∗ (Ξi (x) − ω((1 + r)xi∗ )) (65) = ∆t 1 − (1 + r)xi∗ ≤ ((1 + r)Ξi (x∗ ) − ω((1 + r)xi∗ )) (66) ∆t (1 + r)(1 − (1 + r)xi∗ ) < (Ξi (x∗ ) − ω(xi∗ )) = 0 (67) ∆t |xi (t) − xi∗ | xi (t) − xi∗ Hence, = is strictly decreasing. xi∗ xi∗ ∗ Otherwise, xi = (1 − r)xi . If r < 1, dxi 1 − xi = (Ξi (x) − ω(xi )) (68) dt ∆t 1 − (1 − r)xi∗ = (Ξi (x) − ω((1 − r)xi∗ )) (69) VI. C ONTINUOUS T IME M ODEL ∆t 1 − (1 − r)xi∗ The discrete time model may give an unstable nontrivial ≥ (Ξi ((1 − r)x∗ ) − ω((1 − r)xi∗ )) (70) ∆t fixed point as in (22). However, in the continuous-time model 1 − (1 − r)xi∗ the nontrivial fixed point is globally stable if Ξ and ω satisfy ≥ ((1 − r)Ξi (x∗ ) − ω((1 − r)xi∗ )) (71) ∆t all the properties from (a) to (f). (1 − r)(1 − (1 − r)xi∗ ) Consider a differential equation. > (i(x∗ ) − ω(xi∗ )) = 0 (72) ∆t dxi 1 1 − xi = ((1 − xi )Ξi (x1 , · · · , xn ) − δ xi ) = (Ξi (x) − ω(xi )) |xi (t) − xi∗ | xi∗ − xi (t) dt ∆t ∆t = is strictly decreasing. If r = 1, all (57) xi∗ xi∗ Then, (5) is just the forward Euler method of (57) with ∆t as the entries of x(t) are positive after short time unless x(t) is step size for time. The origin is a trivial equilibrium point of the origin.

The inequality above holds by (18), (19) and (20). It implies that z(k+1) z(k) for every k ∈ N, and z(k) gives a lower e k (x∗ ) = x∗ . bound for P(k). Since z(0) = αx∗ x∗ , z(k) Φ (k) ∞ n {z }k=0 ⊂ [0, 1] has a limit point because it is increasing, and bounded from above. x∗ is the only possible limit point (t) and it goes to of {z(k) }∞ k=0 . The lower bound of P(t) is z ∗ x as t goes to infinity. Both the upper and lower bounds of P(t) go to x∗ which implies that P(t) converges to x∗ . ei ∂Φ The biggest difference of (21) and (52) is that ≥0 ∂xj for any i, j ∈ {1, · · · , n} in (52) while it does not hold for Φ ∂ Φi in (21). The same proof can be applied if ≥ 0 for any ∂xj i, j ∈ {1, · · · , n} in (21).

d |xi (t) − xi∗ | Since < 0 for all i such that dt xi∗ ∗ |xi (t) − xi | d = V (x), V (x(t)) < 0. V (x) is a Lyapunov xi∗ dt function of this system and it completes the proof. We finally remark that, even though the continuous-time and discrete-time models are related through the forward Euler method and that the discrete-time model can be viewed as a discretization of the continuous-time model, it does not mean that continuous-time model is approximation to the true underlying epidemic spread. There are certain applications, such as the interaction of humans over a social network, say, where the discrete-time model appears to be more appropriate. In either case, whether to use a continuoustime model or a discrete-time model (and in the latter case whether to use (1) or the Chakrabarti model) depends on the application at hand. ACKNOWLEDGMENT The authors wish to thank to Subhonmesh Bose, Wei Mao, Matthew Thill and Christos Thrampoulidis for many insightful discussions on the subject. R EFERENCES [1] V. S. Bokharaie, O. Mason and F. Wirth ”Spread of epidemics in time-dependent networks” Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, 2010 [2] B. Bollob´as ”Random Graph”, Academic Press, 1985. [3] D. Chakrabarti, Y. Wang, C. Wang, J. Leskovec and C. Faloutsos ”Epidemic thresholds in real networks” ACM Trans. on Info. and System Security (TISSEC),10(4), 2008. [4] M. Draief ”Epidemic processes on complex networks” Physica A: Statistical Mechanics and its Applications, 2006 [5] E. Estrada, F. Kalala-Mutombo and A. Valverde-Colmeiro ”Epidemic spreading in networks with nonrandom long-range interactions” PHYSICAL REVIEW E 84, 2011 [6] A. Fall, A. Iggidr, G. Sallet and J. J. Tewa ”Epidemiological models and Lyapunov functions” Math. Model. Nat. Phenom. [7] A. Ganesh, L. Massoulie and D. Towsley ”The effect of network topology on the spread of epidemics,” in Proc. IEEE INFOCOM 2005 [8] S. Gomez, A. Arenas, J. Borge-Holthoefer1, S. Meloni and Y. Moreno ”Discrete-time Markov chain approach to contact-based disease spreading in complex networks” EPL(Europhysics Letter). 2010. [9] E. Bodine-Baron, S. Bose, B. Hassibi and A. Wierman ”Minimizing the social cost of an epidemic” Proceedings of ICST Gamenets, 2011. [10] N. Madar, T. Kalisky, R. Cohen, D. Ben-Avraham and S. Havlin ”Immunization and epidemic dynamics in complex networks” The European Physical Journal B - Condensed Matter and Complex Systems, 2004 [11] V. Mieghem, P. Omic and J. Kooij ”Virus spread in networks” IEEE/ACM Trans Network, 2009 [12] R. Pastor-Satorras and A. Vespignani ”Epidemic dynamics and endemic states in complex networks” Physical Review E 63, 2001 [13] R. Pastor-Satorras and A. Vespignani ”Epidemic spreading in scalefree networks” Physical Review E 86, 2001 [14] C. Penga, X. Jina and M. Shic ”Epidemic threshold and immunization on generalized networks” Physica A: Statistical Mechanics and its Applications, 2010 [15] M. Penrose ”Random Geometric Graph”, Oxford University Press, 2003. [16] F. D. Sahneh and C. Scoglio ”Epidemic Spread in Human Networks”, Decision and Control and European Control Conference (CDC-ECC), 2011 [17] Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos ”Epidemic spreading in real networks: An eigenvalue viewpoint” Proc. Symp. Reliable Dist. Systems, 2003.