Susceptible-Infected Epidemics on Evolving Graphs

The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S-I$ connections are broken at rate $\rho$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In \cite{DOMath} the critical value for evoSIR was computed and simulations showed that when $G$ is an Erd\H os-R\'enyi graph with mean degree 5, the system has a discontinuous phase transition, i.e., as the infection rate $\lambda$ decreases to $\lambda_c$, the fraction of individuals infected during the epidemic does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity $\Delta$ determined by the first three moments of the degree distribution, so that the phase transition is discontinuous if $\Delta>0$ and continuous if $\Delta<0$.


Introduction
In the SIR model, individuals are in one of three states: S = susceptible, I = infected, R = removed (cannot be infected). Often this epidemic takes place in a homogeneously mixing population. However, here, we have a graph G that gives the social structure of the population; vertices represent individuals and edges connections between them. model in which edges are deleted instead of rewired. In the evo (or del) version of the model, in order for the infection to be transmitted along an edge, infection must come before any rewiring (or deletion) and before time 1. To compute this probability, note that (i) the probability that infection occurs before rewiring is λ/(λ + ρ) and (ii) the minimum of two independent exponentials with rates λ and ρ is an exponential with rate λ + ρ, so the transmission probability is (1 − e −(λ+ρ) ). (1.1.3) Here the 'r' subscript is for "rewire." By a standard coupling argument one can show that evoSI dominates delSI.

Lemma 1.2.
For fixed parameters, there exists a coupling of evoSI and delSI so that there are no fewer infections in the delSI model than in evoSI. Same is true if we replace SI by SIR.
The next result, Theorem 1 in [14], shows that evoSIR has the same critical value as delSIR, and in the subcritical case the expected cluster sizes are the same. Theorem 1.3. The critical value λ c for a large epidemic in fixed infection time delSIR or evoSIR epidemic with rewiring is given by the solution of µτ f r (λ) = 1. Moreover, if λ < λ c , then the ratio of the expected epidemic size in delSIR to the size in evoSIR converges to 1 as the number of vertices goes to ∞.
The formula for the critical value is easily seen to be correct for the delSIR since, by the reasoning above, there is a large epidemic if and only if the reduced graph in which edges are retained with probability τ f r has a giant component. From Lemma 1.2 one can see that the delSIR model has a larger (≥) critical value than evoSIR. Thus, one only has to prove the reverse inequality. Intuitively, the equality of the two critical values holds because a subcritical delSIR epidemic dies out quickly, so it is unlikely that rewirings will influence the outcome.
When n is large, the degree distribution, which is Binomial(n − 1, µ/n), is approximately Poisson with mean µ. Due to Poisson thinning, the number of new infections directly caused by one I in delSIR in an otherwise susceptible population is asymptotically Poisson with mean µτ f r , and hence has limiting generating function of the distribution Poisson(µτ f ) G 1 (z) = exp(−µτ f r (1 − z)). (1.1.4) The following result, Theorem 2 in [14], identifies the probability of a large outbreak.
Theorem 1.4. If z 0 < 1 is the fixed point of G 1 (z), then 1 − z 0 gives the probability of a large delSIR or evoSIR outbreak.
In the case of the delSIR model, 1 − z 0 is the fraction of individuals infected in a large epidemic. It is easy to see that this proportion goes to 0 at the critical value µ c = 1/τ f r = 1.
The next simulation suggests that this is not true in the case of evoSIR.

Britton et al. [3, 16]
As the authors of [14] were finishing up the writing of their paper, they learned of two papers by Britton and collaborators that study epidemics on evolving graphs with exponential infection times. [3] studies a one parameter family of models (SIR-ω) that interpolates between delSIR and evoSIR. To facilitate later referencing we attach labels to the next two descriptions. Model 1. SIR-ω epidemic. In this model, an infected individual infects each neighbor at rate λ, and recovers at rate γ. A susceptible individual drops its connection to an  [14] that turned out to be inaccurate. The top curve comes from simulating evoSIR.
infected individual at rate ω. The edge is rewired with probability α and dropped with probability 1 − α. Since evoSIR (α = 1) and delSIR (α = 0) have the same critical values and survival probability it follows that this holds for all 0 ≤ α ≤ 1 since by a coupling argument the final epidemic size of the case 0 < α < 1 can be sandwiched between delSIR and evoSIR.
These epidemics take place on graphs generated by Model 2. Configuration model. Given a nonnegative integer n and a positive integer valued random variable D, take n i.i.d. copies D 1 , . . . , D n of D. If the sum n i=1 D i is odd, then we replace D n by D n + 1. We then construct a graph G on n vertices as follows. We attach D 1 , . . . , D n half-edges to vertices 1, 2 . . . , n, respectively and then pair these half-edges uniformly at random to form a graph. We call this random graph the configuration model on n vertices with degree distribution D and denote it by CM(n, D). We assume D has finite second moment so that the resulting graph will be a simple graph with nonvanishing probability as n → ∞. See Theorem 3.1.2 in [7]. We refer readers to [22,Chapter 7], [24,Chapters 4 and 7] and [23,Chapter 2] for more details on the configuration model. Remark 1.5. Throughout the paper, unless otherwise specified, we always consider the annealed probability measure with respect to the configuration model. In other words the randomness is taken over both the degrees D 1 , . . . , D n and the construction of CM(n, D) based on the degrees. Britton, Juher, and Saldana [3] studied the initial phase of the epidemic starting with one infected at vertex x (chosen uniformly at random from all vertices) using a branching process approximation. Let Z n m be the number of vertices at distance m from x in the graph CM(n, D). For any fixed k ∈ N, {Z n m , 0 ≤ m ≤ k} converges to the following two-phase branching process {Z m , 0 ≤ m ≤ k}. The number of children in the first generation has the distribution D while subsequent generations have the distribution D * − 1 where D * is the size-biased degree distribution P(D * = j) = jp j m 1 , j ≥ 0.
Here p j = P(D = j) and m 1 = E(D) is the mean of D. Later we will also use m i := E(D i ) to denote the i-th moment of D for i ≥ 1. This follows from the construction of the configuration model: x connects to other vertices with probability proportional to their degrees so individuals in generations m ≥ 1 have the D * − 1 children instead of D. The '-1' is because one edge is used in making the connection from x. Before moving on to epidemics on the configuration model, we note that if then the generating function of D * − 1 is (1.

2.2)
In the SIR-ω model, the probability that an infection will cross an S − I edge is τ = λ λ + γ + ω .
One can see from this description that the limiting branching processZ m will have positive survival probability if Correspondingly, there will be a large epidemic in the SIR-ω model if This follows from results on percolation in random graphs (see [9] and [12]). Thus for fixed values of γ and ω λ c = (γ + ω) m 1 m 2 − 2m 1 .
(1.2.4) When γ = 0 and ω = ρ which is the SI-ω model in our notation, The critical values of delSI and evoSI only depends on the ratio ρ/λ, so it is natural to define a parameter (this α is different from the α used in the definition of SIR-ω model) α = ρm 1 /λ (1.2.6) that has α c = m 2 − 2m 1 . We will only consider the del and evo endpoints of the one parameter family of models SIR-ω, so after the discussion of previous work is completed, there should be no confusion between our α and theirs.
Work of Leung, Ball, Sirl, and Britton [16] demonstrated the paradoxical fact that individual preventative measures may lead to a larger final size of the epidemic. They proved this rigorously for SI epidemics on the configuration model with two degrees and conducted simulation studies for many social networks. Note that in Figure 2 (taken from Figure 1 of [16]) the final size increases with the rewirng rate when the rewiring rate is small. This simulation does not suggest that the phase transition was discontinuous.
On the other hand, Figure 3 from [14] gives a similar simulation that clearly shows the discontinuity.  [14] that turned out to be inaccurate. The top curve comes from simulating evoSIR.

Ball and Britton [2]
Ball and Britton [2] analyzed the evoSIR and evoSI epidemics on Erdős-Rényi graphs. Their construction uses properties that are special to that case. See [2,Section 2.3] for details and also [4,21] for earlier examples of the use of this construction. They solved the case of the SI-ω model on Erdős-Rényi graphs completely, but for the SIR-ω model there is a gap between the necessary and sufficient condition for a discontinuous phase transition. See (1.3

.3) and the comment after it.
To prove results about the epidemics on an Erdős-Rényi random graph with mean degree µ they first consider a tree in which each vertex has a Poisson(µ) number of descendants and develop a branching process approximation for the SIR-ω epidemic in which infections (births) cross an edge with probability λ/(λ + γ + ω). Let I(t) be the total number of infected individuals, I E (t) be the number of infectious edges, and T (t) be the total progeny in the branching process on the tree. Let I n (t), I n E (t) and T n (t) be the corresponding quantities for an Erdős-Rényi(n, µ/n) random graph on n vertices where initially a randomly chosen vertex is infected. They show in their Theorem 2.1 that if t n = inf{t : T (t) ≥ log n} then the two systems can be defined on the same space so that sup 0≤t≤tn |(I n (t), I n E (t), T n (t)) − (I(t), I E (t), T (t))| P − → 0 as n → ∞. Let S n (t) be the number of susceptibles at time t and W n (t) be the number of susceptible-susceptible edges created by rewiring by time t and let X n (t) = (S n (t), I n (t), I n E (t), W n (t)). Let x(t) = (s(t), i(t), i E (t), w(t)) be the solution of the ODE Here α is the probability that an edge is rewired as in the definition of SIR-ω model.
Theorem 2.2 in [2], which is proved using results of Darling and Norris [5], shows that as n → ∞, provided that I n (t)/n → i(0) > 0. It is interesting to note, see their Section 3, that the ODE in (1.3.1) is closely related to the "pair approximation" for SIR-ω model. To explain the phrase in quotes, we note that "mean-field equations" come from pretending that the states of site are independent; the pair approximation from assuming it is a Markov chain. In practice, this approach means that probabilities involving three sites are reduced to probabilities involving 1 and 2 sites using a conditional independence property. For the details of the computation see Chapter 7 in [1].
Letting T n = n − S n (∞) (which is the final size of the epidemic) they make Conjecture 2.1 that one can interchange two limits n → ∞ and t → ∞ to conclude To formulate a result that is independent of the validity of the conjecture they let be the solution to the ODE when
For the case of the delSIR model (α = 0, ω = ρ and γ = 1) studied in [14], the transition is always continuous. This follows from Theorem 2.3 since 1 > −ρ. From the last calculation we see that the phase transition is always continuous if α < 1/2.
In the case of the SI-ω model they show (see Theorem 2.6 of [2]) that if µ > 1 and ω and α are held fixed then the phase transition is discontinuous if and only if α > 1/3 and µ > 3α/(3α − 1). When α = 1 this is µ > 3/2 which is the condition in Example 1.
See their paper for a precise statement. Remark 2.4 in [2] states the conjecture that the 3's in the first condition should be 2's.
To give the value of τ 0 (µ, α), we need some notations. For µ > 1 and α ∈ [0, 1] let , Ball and Britton also made connections of their paper with our paper (as well as an earlier version of this paper) in [2,Section 4]. In particular, they showed in their Figure   5 that in the case µ = 2 their predicted final size τ 0 (2, 1) agrees with simulation results well.

Statement of our main results
From now on the reader can forget about the meaning of notations used by Ball and Britton. We fix ρ, the rewiring rate, and vary λ. We let α = ρm 1 /λ. In view of the definition of ∆ in (1.4.1), the natural assumption is E(D 3 ) < ∞. Some of our results can be proved under this assumption, while some need something a little stronger. Specifically, we need finite fifth moment to prove (1.4.2). To simplify things we assume (ii) When α < α c , which is the supercritical case, the probability of a large epidemic is the same in the two models, which is equal to the survival probability q(λ) of the two-stage branching processZ m defined in Section 1.2 (with the τ there equal to λ/(λ + ρ) in our notation).
The proof of Theorem 1.6 that we give in Section 2 is very similar to one for Theorem 1.3 given in [14] for Erdős-Rényi random graphs. Here the fact that we have only E(D 5 ) < ∞ rather than exponential upper bounds on P(D ≥ k) changes some of the estimates.
Here and in what follows, formulas are sometimes easier to evaluate if we use the "factorial moments" µ k = E[D(D − 1) · · · (D − k + 1)], since these can be computed from the k-th derivative of the generating function. To translate between the two notations: Our next result gives an almost sufficient and necessary condition for the discontinuous phase transition of evoSI. We use the word 'almost' since the case ∆ = 0 is not treated here.

4.3)
To see what this result says we consider some examples.
so the phase transition is discontinuous if p < 1/2.
Our last example concerns the configuration model generated from Poisson distribution: Example 1.10. Poisson(µ). The factorial moments µ k = µ k , so the critical value α c = µ 2 − µ 1 = µ 2 − µ, which is positive if µ > 1. This condition is natural since if µ < 1 then there is no giant component in the graph and a large epidemic is impossible.
so the phase transition for evoSI is discontinuous if µ > 3/2, which is the result given in [2]. Remark 1.11. We believe that the result of Example 1.10 also holds for Erdős-Rényi(n, µ/n).
To prove this rigorously, one first has to prove a quenched version of Theorem 1.7 (i.e., showing that (1.4.2) and (1.4.3) hold with high probability over any degree sequence D 1 , . . . , D n (that are not necessarily i.i.d.) such that the k-th factorial moment of the empirical distribution converges to µ k for any k ≥ 0). We believe that this can be shown using the same ideas in the proof of Theorem 1.7. Then one can transfer results for the configuration model to Erdős-Rényi(n, µ/n) using [22,Theorems 7.18 and 7.19], which says that conditionally on having the same degrees, the random graphs generated from these two models have the same distribution.
As a notational note, in this paper we will use C, C 1 , C 2 , · · · to denote various constants whose specific values might change from line to line. Occasionally when we have an important constant we will number it by the formula it first appeared in, e.g., C 2.3.2 below.

Sketch of Proof of Theorem 1.7
The proof of Theorem 1.7 is done by constructing auxiliary models that are upper/lower bounds for evoSI. We introduce a process which we call avoSI (avo is short for avoiding infection) in Section 3.1 and prove that the final set of infected sites in avoSI stochastically dominates evoSI. We also construct a lower bounding process which we call AB-avoSI in Section 4.1, where we prove that the final set of infected sites in evoSI stochastically dominates AB-avoSI. The AB in the name comes from the two counters associated with half-edges that prevent transmission of infections along S − I edges created by I − I rewirings.
The starting point to analyze evoSI via avoSI and AB-avoSI is the following Lemma 1.12. Let q(λ) be the survival probability for the two-phase branching process {Z m , m ≥ 0} introduced in Section 1.2. Recall that the individual in the first generation has offspring distribution Binormial(D, λ/(λ + ρ)) while later generations have offspring distribution Binormial(D * − 1, λ/(λ + ρ)) where D * is the size-biased version of D, of epidemic2. In fact, we will prove this chain of comparisons in Lemmas 3.2, 4.1 and 4.2, respectively. It remains to show that avoSI and delSI has the same critical value and probability of a large outbreak. This is proved in Lemma 3.3.
Below we will use λ c and α c = ρλ c /m 1 to denote the critical value. Recall the definition of the generating function G in (1.2.1). Consider a function f defined by (1.5.1) Consider the avoSI epidemic on the configuration model CM(n, D) with one uniformly randomly chosen vertex initially infected. Suppose α < α c so that we are in the supercritical regime. Let I ∞ be the final epidemic size. Set If we suppose (⋆) either σ = 0 or 0 < σ < 1 and there is a δ > 0 so that f < 0 on (σ − δ, σ), then for any ϵ > 0, Though ν does not give the correct final size of the evoSI epidemic, the formula for f (w) is accurate enough for w near 1 to identify when the phase transition is continuous.
Theorem 1.14. Consider the avoSI epidemic on the configuration model CM(n, D) and let I ∞ be the final epidemic size. Set (1.

5.4)
If ∆ < 0, then there a continuous phase transition. For any ϵ > 0, there exists some δ > 0, so that lim n→∞ P( I ∞ /n > ϵ) = 0 for α c − δ < α < α c . (1.5.5) We can show that ∆ > 0 implies that there is a discontinuous phase transition in avoSI, but that result does not help us prove Theorem 1.7. To get Theorem 1.14 from Theorem 1.13 we compute, see Section 3.6, that When ∆ > 0, as w decreases from 1 the curve of f turns up, and σ stays bounded away from 0. When ∆ < 0, the curve of f turns down, and σ converges to 1 as α → α c . See Figure 4.  Figure 4: The behavior of f (w) near 1 with respect to different α ′ s for the Erdős-Rényi graph. In the top graph µ = 1.4, which has ∆ < 0. α c = µ 2 − µ = .56. Notice that as α increases to 0.56 the intersection with the x axis tends to 1, so the transition is continuous. In the bottom graph µ = 3, which has ∆ > 0. α c = µ 2 − µ = 6. Notice that when α ≤ α c , f (w) > 0 for w ∈ [0.9, 1), so σ is bounded away from 1.

Sketch of Proof of Theorem 1.13
To begin to explain the ideas behind the analysis of epidemics on evolving graphs we need to recall some history. Volz [27] was the first to derive a limiting ODE system for an SIR epidemic on a (static) graph generated by the configuration model. Miller [18] later simplified the derivation to produce a single ODE. The results of Volz and Miller were based on heuristic computations, but later their conclusion was made rigorous by Decreusfond et al [6] assuming E(D 5 ) < ∞.
Janson, Lukzak, and Windridge [13] proved the result under more natural assumptions. They studied the epidemic on the graph by revealing its edges dynamically while the epidemic spreads. Recall that the configuration is constructed using half-edges. The authors in [13] call a half-edge free if it has not yet been paired with another half-edge. They call a half-edge susceptible, infected or removed according to the state of its vertex. To modify their construction to include rewiring we add the third bullet below. Hereafter we use"randomly chosen" and "at random" to mean that the distribution of the choice is uniform over the set of possibilities.
• Each free infected half-edge chooses a free half-edge at rate λ. Together the pair forms an edge and is removed from the collection of half-edges. If the pairing is with a susceptible half-edge then its vertex becomes infected and all its edges become infected half-edges.
• Infected vertices recover and enter the removed state at rate 1.
• Each infected half-edge gets removed from the vertex that it is attached to at rate ρ and immediately becomes re-attached to a randomly chosen vertex.
To analyze the avoSI model we follow the approach in Janson, Luczak, and Windridge [13] and construct the graph as we run the infection process. The construction of this process and its coupling to evoSI are described in Section 3.1. Initially the graph consists of half-edges connected to vertices, as in the configuration model construction before the half-edges are paired. Let X t be the total number of half-edges at time t and let X I,t be the number of half-edges that are attached to infected vertices and let S t,k be the number of susceptible vertices with k half-edges at time t. The evolution of S t,k in avoSI is given by, see (3.2.1), where M t,k is a martingale and we have returned to using λ as the infection rate and ρ as the rewiring rate. Following [13] we time-change the process by multiplying the original transition rates by ( X t − 1)/(λ X I,t ). Let X t be the number of half-edges at time t in the time changed process, and let X S,t be the number of half-edges that are attached to susceptible vertices. Using S t,k for the time-changed process the new dynamics are, see (3.2.3), (1.6.1) Note that, thanks to the time change, the number of infected half-edges X I,t no longer appears in the equation. Let γ n be the first time there are no infected half-edges. Let w(t) = exp(−t) and m 1 = E(D). The key to the proof of Theorem 1.13 is to show (1.6.3) From the results above, we see that . (1.6.4) The logarithm of the right-hand side is f (w). Under assumption (⋆), σ = sup{w : 0 < w < 1, f (w) = 0} gives the time z = − log(σ) at which the infection dies out in the time-changed process and ν defined in (1.5.3) gives the fraction of sites which have been infected.
There are four steps in the proof of (1.6.2): • In Section 3.2 we show that for each fixed k ∈ N, {S t,k /n, t ≥ 0} n≥1 is a tight sequence of processes.
• In Section 3.3 we show that any subsequential limit satisfies a system of differential equations (3.3.4) that has a unique solutions t,k , so S t,k /n →s t,k .
• Section 3.4 we deal with the technicality of showing that the limit of ∞ k=0 kS t,k /n is the sum of the limits ∞ k=0 ks t,k . • In Section 3.5 we complete the proof by establishing the formulas for σ and ν.

Sketch of Proof of Theorem 1.15
In the AB-avoSI model, each half-edge i has two indices A(i, t) and B(i, t).
• The infection index A(i, t) = 0 if i has not been infected by time t. If i first become an infected half-edge at time s, then we set A(i, t) = s for all t ≥ s. • The rewiring index B(i, t) = 0 if i has not been rewired by time t. If i gets rewired at time s, then we update the value of B(i, s) to be s, regardless of whether i has been rewired before or not. B(i, ·) remains constant between consecutive rewirings.
Suppose an infected half-edge i pairs with a susceptible half-edge j at tine t, then (in the AB-avoSI model) i will transmit an infection to j if and only if A(i, t) > B(j, t). See Section 4.1 for more details about the AB-avoSI model and its relationship to evoSI. Leť S t,k be the number of susceptible vertices with k half-edges at time t and set Here I(i, t) is an indicator function such that I(i, t) = 1 if half-edge i is an infected half-edge at time t (see the first paragraph of Section 4.2 for the definitions of the notations I(i, t), S(i, t), S(i, k, t), D(j, t) appearing below). As in the avoSI model we make a time-change by multiplying the original transition rates by (X t − 1)/(λX I,t ). Using a hat to denote the quantities after the time-change in the AB-avoSI model, we have that, for all k ≥ 0, where M t,k is a martingale. See equation (4.2.2). This system of equations is not solvable but we can expand in powers of t to study the time-changed system for small t. If we let X t , X I,t , S t be the number of half-edges, the number of infected half-edges and the number of susceptible vertices, respectively, then we have where M t is a martingale. See equation (4.3.12). Define By expanding S t,k around t = 0 up to the second order, we get, for any ϵ > 0, λ close to λ c and t 0 close to 0, Remark 1. 16. In the case of avoSI, we have that, for n large, . By expanding f up to the second order and using (1.5.6) along with the fact that exp(−t) = 1 − t + t 2 /2 + o(t 2 ), we get for small t and λ close to λ c . In the case of AB-avoSI, see (4.5.12), we have, as a lower bound when t > ϵ (the ϵ 2 and ϵ 6 here are some small numbers depending on ϵ). The two expansions do not match but both linear terms vanish at λ c and the quadratic terms have the same sign so this is good enough.
The proof of Theorem 1.15 is organized into five steps: • In Section 4.1 we define the AB-avoSI process and prove that evoSI stochastically dominates AB-avoSI. • In Section 4.2 we derive basic moment estimates for various quantities that will prepare us for later proofs. See Lemma 4.3.
• In Section 4.3 we give rough upper and lower bounds for I t and X I,t involving the integral of E(t). See Lemma 4.4. We also give an easy upper bound for E(t) in (4.3.25).
• In Section 4.4 we decompose E(t) into two parts (see (4.4.1)) and give refined bounds for each part. See Lemmas 4.5 and 4.6. • In Section 4.5 we combine our estimates to complete the proof.
2 Proof of Theorem 1.6

Coupling of evoSI and delSI
We first prove Lemma 1.2 before proving Theorem 1.6. We define three sequences of random variables, which will serve as the joint randomness to couple evoSI and delSI: • Let T e,ℓ , ℓ ≥ 1 be independent exponential random variables with mean 1/λ. • Let R e,ℓ , ℓ ≥ 1 be independent exponential random variables with mean 1/ρ. • Let U e,ℓ , ℓ ≥ 1 be independent random variables chosen uniformly at random from all vertices.

Construction of evoSI.
We define three sets of edges in evoSI at time t: • Active edges, denoted by E a t , are the edges at time t that connect an infected vertex and a susceptible vertex.
• Uninfected edges, denoted by E 0 t , connect two susceptible vertices.
• Inactive edges, denoted by E i t , have both ends infected. Once an edge becomes inactive it remains inactive forever.
The three sets form a partition of all edges.
The three set-valued processes just defined are right-continuous pure jump processes.
At time 0 we randomly choose a vertex u 0 to be infected. E a 0 consists of the edges with one endpoint at u 0 . E 0 0 is the collection of all edges in the graph minus the set E a 0 . E i 0 = ∅. We will consider the corresponding sets for delSI, but they will be denoted by D t to avoid confusion.
For each undirected edge e, let τ e e,ℓ be the ℓ-th time the edge becomes active (the superscript 'e' is short for evo). To make it easier to describe the dynamics, suppose that at time τ e e,ℓ we have e = {x e,ℓ , y e,ℓ } with x e,ℓ infected and y e,ℓ susceptible.
• Let T e,ℓ , ℓ ≥ 1 be the time between τ e,ℓ and the infection of y e,ℓ by x e,ℓ .
• Let R e,ℓ , ℓ ≥ 1 be the time between τ e,ℓ and the time when y e,ℓ breaks its connection to x e,ℓ and rewires. • Let U e,ℓ be the vertex that y e,ℓ connects to at time τ e e,ℓ + R e,ℓ (if rewiring occurs).

Initial step.
To simplify writing formulas, let S e,ℓ = min{T e,ℓ , R e,ℓ }. (i) If R e0,i,1 < T e0,i,1 , then at time J e 1 vertex y i breaks its connection with u 0 and rewires On the initial step this will hold unless U e0,i,1 = u 0 in which case nothing has changed.

Induction step.
For any active edges e present at time t, let L e (e, t) = sup{ℓ : τ e e,ℓ ≤ t} and let V e (e, t) = τ e e,L e (e,t) + S e,L e (e,t) be the time of the next event (infection or rewiring) to affect edge e. Again, the superscripts 'e' in L e (e, t) and V e (e, t) imply that these quantities are for the evoSI model.  Black dots mark infected sites.

Construction of delSI.
There is no rewiring in delSI, each edge will become active at most once. Thus for each undirected edge e we only need two exponential random variables T e,1 and R e,1 , defined in the beginning of Section 2.1. This allows us to couple evoSI and delSI. Also, we use D 0 t , D a t and D i Initial step. At time 0, a randomly chosen vertex u 0 is infected. The edges N 0 (u 0 , 0) = {e 0,1 , . . . , e 0,k } are added to the list of active edges. We have D a 0 = {e 0,1 , . . . , e 0,k }. Suppose e j connects u 0 and y j . At time the first event occurs. The superscript 'd' stands for delSI and S e,1 = min{T e,1 , R e,1 }, as defined in (2.1.1). Let i be the index that achieves the minimum.
(i) If R e0,i,1 < T e0,i,1 , then at time J a 1 the edge e 0,i is removed from the graph (and hence also from the set D a J d 1 ).
(ii) T e0,i,1 < R e0,i,1 then at time J a 1 vertex y i is infected by x i . We move e 0,i to D i J d

1
. We Induction step. For any active edge e at time t, let τ d e,1 be the first time that e becomes active in the delSI process. We also let  • We move all edges e ′ in N 0 . Since y(e m ) has just become infected, the other end of e ′ must be susceptible at time J d m+1 • We move all edges e ′′ in N a . Since y(e m ) has just become infected, (i) the other end of e ′′ must be infected at time J d m+1 , and (ii) e ′′ cannot have been inactive earlier.
We now prove by induction that avoSI dominates evoSI.
Lemma 2.1. All vertices infected in delSI are also infected in evoSI and are infected earlier in avoSI than evoSI.
Proof. The induction hypothesis holds for the first vertex since u 0 is infected at time 0 in both evoSI and delSI. Suppose the induction holds up to the k-st infected vertex in delSI. Assume at time t, y becomes the (k + 1)-st infected vertex in delSI and y is infected by vertex x through edge e. We see from the construction of delSI that this implies T e,1 < R e,1 . Suppose x was infected at time s < t in delSI. The induction hypothesis implies that x has also been infected in evoSI at a time s ′ ≤ s. There are two possible cases for y in evoSI: • y has already become infected by time s ′ + T e,1 .
• y was still susceptible right before s ′ + T e,1 . In this case, y will be infected at time s ′ + T e,1 ≤ s + T e,1 = t.
In either case x has been infected by time t in evoSI. This completes the induction step and thus proves Lemma 1.2.

The infected sites in delSI
In the introduction we have noted that delSI is equivalent to independent bond percolation. That is, we keep each edge independently with probability λ/(λ + ρ) and find the component containing the initially infected vertex (say, vertex 1). To compute the size of the delSI epidemic starting from vertex 1, we apply a standard algorithm, see e.g., [17], for computing the size of the component containing 1 in the reduced graph in which edges have independently been deleted with probability ρ/(λ + ρ). We call this the exploration process of delSI. At step 0 the active set A 0 = {1}, the unexplored set U 0 = {2, . . . n}, and the removed set R 0 = ∅. Here removed means these sites are no longer needed in the computation. In the SI model sites never enter the removed state, Let η i,j = η j,i = 1 if there is an edge connecting i and j in the reduced graph. If η i,j = 1 an infection at i is transmitted to j. At step t if A t ̸ = ∅ we pick an i t ∈ A t and update the sets as follows.
When A t = ∅ we have found the cluster containing 1 in the reduced graph, which will be the final set of infected sites in the SI model. The exploration process of the configuration model can be similarly defined (just without deletion of edges) and we let J t be the set of active sites at step t in this exploration. We set R t = |R t+1 | = t + 1, A t = |A t+1 | and J t = |J t+1 |. We make a time shift so that A t and J t can be coupled with two random walks with i.i.d. increments (see Lemma 2.2 below). We now study the exploration process of the configuration model itself as well as the delSI process on such graph. Let ψ 0 have generating function G defined in (1.2.1) and ζ 0 have a Binomial(D, λ/(λ + ρ)) distribution whose generating function is denoted by G ρ .
The two generating functions are related by Recall the definition of the generating function G in (1.2.2). We can similarly define G ρ . Let {χ i , i ≥ 1} and {ξ i , i ≥ 1} be independent random variables with generating functions G and G ρ , respectively. Define two random walks (for integer-valued t): . We defineS t in a similar way.
Note thatW t andS t can be viewed as the exploration processes of two-phase branching process Z m andZ m (both are defined in Section 1.2), respectively. The proof of Lemma 2.2 is deferred to the end of this section. For the rest of this section we will always work on the event and hence assume that A t and J t have independent increments until they hit 0.

Proof of Theorem 1.6(i)
The formula for the critical value of delSI follows from standard results on percolation in random graphs. Note that in delSI each edge is kept with probability λ/(λ + ρ). Using [12,Theorem 3.9] we see that Recall that Lemma 1.2 shows that the final set of infected individuals in delSI is contained in the analogous set for evoSI with the same parameters so To prove that the two are equal we will show that if λ < λ c (delSI) then evoSI dies out, i.e., infects only a vanishing portion of the total population as n → ∞.
To compare the two evolutions, we will first run the delSI epidemic to completion. Once this is done we will randomly rewire the edges deleted in delSI. If the rewiring creates a new infection in evoSI, then we have to continue to run the process. If not, then the infected sites in the two processes are the same. Let R be the set of sites that are eventually infected in delSI, and let R ′ be the set of eventually infected sites in evoSI. Let R ′ = |R ′ | and R = |R|.
To get started we use a result of Janson [11] about graphs with specified degree distributions. He works in the set-up introduced by Molloy-Reed [19,20] where the degree sequence d n i , 1 ≤ i ≤ n is specified and one assumes only that limiting moments exist as well some other technical assumptions that are satisfied in our case (for a more recent example see [13]): The next result is Theorem 1.1 in [11]. The µ and θ in this theorem have the same meaning as (2.3.1). The "whp" below is short for with high probability, and means that the probability the inequality holds tends to 1 as n → ∞. Let D n = d n u0 where u 0 is randomly chosen from {1, 2, . . . , n}.
Then there is a constant C 2.3.2 which depends on C so that the largest component C 1 has Theorem 2.3 can also be applied to the setting where the degrees are random rather than deterministic. We have assumed that in the original graph It follows that Let N d be the number of deleted edges in the exploration process of delSI. One vertex is removed from the construction on each step, so whp the number of steps is ≤ r 0 := C 2.3.2 n 1/3 . Note that r 0 ≪ n 1/3 log n so we can assume J t has independent increments by Lemma 2.2. It follows that where we recall that the χ i are independent with the distribution D * − 1. The inequality comes from the fact that we are counting all the edges even if they are not deleted.
A similar argument shows that Since a > 1/3, when n is large we can upper bound r 0 Eχ 1 + Eψ 0 + r 0 + n a and r 0 Eξ 1 + Eζ 0 + r 0 + n a by 2n a .
At step r 0 we use random variables independent of delSI to randomly rewire the deleted edges. Let Y be the number of edges deleted up to time r 0 that rewire to the set A r0+1 ∪ R r0+1 . By construction where ≡ indicates that the last equality definesȲ . From this we get completes the proof of (i). To prepare for the proof of (ii) note that the conclusion (the set of infected sites coincide in delSI and evoSI) holds as long as the number of steps is smaller than Cn 1/3 even if the epidemic is supercritical.

Proof of Theorem 1.6(ii)
We need the following ingredient in the proof.

Lemma 2.4.
There is a γ > 0 so that Hence if we take γ = E(ξ 1 )/2 then we have Let B d and B e be the events that there is a large epidemic in delSI and evoSI respectively. We now use Lemma 2.4 to show the difference between the probabilities of these two events vanishes asymptotically.
We note that • P(F 1 ) converges to 0 by Lemma 2.4. • P(B d |F 2 ) converges to 1 by the same argument as the proof of Theorem 2.9(c) in [13] (see page 750-752 in [13]).
Therefore we have P(B d ) − P(F 2 ) → 0. As for P(B e ), we note that by the remark at the end of the proof of (i) one has P(A log n = A ′ log n ) → 1 where A ′ log n is the number of active sites in evoSI at step log n. Using the decomposition P( This completes the proof of Lemma 2.5. To compute the limit of P(A log n > 0), note that due to Lemma 2.2, A t can be coupled with the exploration process of the two-phase branching processZ m defined in Section 1.2. Therefore we have (recall that q(λ) is the survival probability ofZ m ) By (2.4.1) we see that P(F 2 ) → q(λ) as well. This implies that both P(B e ) and P(B d ) converge to q(λ) as n → ∞ and completes the proof of Theorem 1.6(ii). Note that using the fact that delSI is equivalent to independent bond percolation, the statement P(B d ) → q(λ) also follows from standard results on percolation in random graphs. See, e.g., [12,Theorem 3.9]. It remains to prove Lemma 2.2 to complete the proof of Theorem 1.6.
Proof of Lemma 2.2. We only prove equation (2.2.2) since the other one follows from (2.2.2) and the fact that delSI is equivalent to percolation with edge retaining probability λ/(λ + ρ). The proof consists of two steps. First, we define an empirical version of W t . Let D 1 , . . . , D n be i.i.d. random variables sampled from the distribution of D. Given a sample of D 1 , . . . , D n , let ψ n 0 be sampled from the (random) distribution In other words, W n t is a random walk in the random environment given by D 1 , . . . , D n .
We defineW n t to be W n t∧τ ′ where τ ′ is the first time W n t hits zero. Note that the condition Using Lemma 2.12 in [23], we see that (recall that J t is the exploration process of the configuration model starting from a uniformly randomly chosen vertex) • J t can be coupled withW n t with high probability up to time n 3/8 / log n. • Whp the subgraph obtained by exploring the neighborhoods of n 3/8 / log n vertices is a tree.
To prove (2.2.2) it remains to show that one can couple W n t and W t up to step n 1/3 log n.
To this end, we use the characterization of total variation distance in terms of optimal coupling. It is well known that for any two random variables X and Y , Here the superscript D n indicates that we are considering the quenched law of ψ n 0 and ξ n 0 . Since conditionally on D n , ψ n 0 and χ n r , r ≥ 1 are all independent, We need the following lemma to control D n . Recall that we let p k = P(D = k) and m 1 = E(D). Also recall that we assume E(D 5 ) < ∞. Lemma 2.6. For any ϵ > 0, Let N k be the cardinality of the set {i : Proof For the second inequality, we note that (2.4.8) (2.4.9) Lemma 2.6 and equation (2.4.2) imply that P(Ω n ) → 1 by the union bound. We now control d Dn TV (χ n 1 , χ 1 ) on Ω n . For k ≥ n 1/4 log n, we have P n (χ n 1 = k) = 0 on Ω n almost surely. We also have k≥n 1/4 log n P(χ 1 = k) ≤ n −1 E(D 4 ) ≤ Cn −1 . Hence we have (2.4.10) For 0 ≤ k ≤ n 1/4 log n, using the definition of Ω n we get, for n large, where we have used the fact that p k ≤ Ck −4 since E(D 4 ) < ∞ in the last step. Summing the last line of (2.4.11) over k from 0 to n 1/4 log n and using the facts

avoSI
As mentioned in the introduction, in this we will construct a model that serves as an upper bound for evoSI. We first introduce a model C-evoSI where 'C' stands for 'coupled' which means we couple the structure of the graph with the epidemic. The 'avo' stands for avoiding infection.
The C-avoSI process is constructed as follows. First recall that in the construction of the configuration model each vertex is assigned a random number of half-edges initially. In the beginning all half-edges attached to the n vertices are unpaired. The half-edges attached to infected nodes are called infected half-edges and those attached to susceptible nodes are susceptible half-edges. Recall that"randomly chosen" and "at random" mean that the distribution of the choice is uniform over the set of possibilities.
• At rate λ each infected half-edge pairs with a randomly chosen half-edge in the pool of all half-edges excluding itself. If the vertex y associated with that half-edge is susceptible then it becomes infected. Note that if vertex y changes from state S to I then all half-edges attached to y become infected half-edges.
• Each infected half-edge gets removed from the vertex that it is attached to at rate ρ and immediately becomes re-attached to a randomly chosen vertex in the pool of all vertices.
For the purpose of comparisons it is convenient to give a reformulation of C-avoSI where the graph has been constructed before the epidemic. We call this process avoSI.
To describe this process we define the notion of 'stable edge'. We say that an edge between two vertices x and y is stable if one of the following conditions hold: • Both x and y are in state S.
• Either x or y has sent an infection to the other one through this edge.
We say that an edge is unstable if both conditions fail. Note that an S − I pair is necessarily unstable since the vertex in state I has not sent an infection to the vertex in state S. We define the avoSI process as follows: • Each infected vertex sends infections to its neighbors at rate λ. If the neighbor has already been infected then nothing changes. Once a vertex receives an infection, it stays infected forever.
• A vertex in state S will break its connection with a vertex in state I at rate ρ and rewire to another randomly chosen vertex. The events for different S − I connections are independent.
• For every unstable I − I edge, each of the two I ′ s will rewire at rate ρ to another uniformly chosen vertex. (This also explains why we call such edges 'unstable', since they may evolve.) Lemma 3.1. The C-avoSI process and avoSI process running on a configuration model have the same law in terms of the evolution of the set of infected vertices.
Proof of Lemma 3.1. It suffices to construct a graph G that has the law of the configuration model such that the set of infected vertices in C-avoSI evolves in the same way as that of the avoSI with initial underlying graph being G. Given an outcome of C-avoSI, the graph G can be constructed as follows. Let H be the collection of vertices and half-edges.
• We assign a unique label to each half-edge in C-avoSI and correspondingly label the half-edges in H.
• Whenever two half-edges combine into one edge in C-evoSI process we pair the two half-edges with the same labels in H. It is clear that the pairings of half-edges are done at random. We pair the remaining half-edges at random after there is no infected half-edge in the system. This forms the graph G. Since the pairings of all half-edges are at random, we deduce that G itself has the law of CM(n, D).
• Whenever a pairing occurs in C-avoSI, an infected vertex has sent an infection to one of its neighbors(s) in avoSI.
• Whenever an infected half-edge h attached to x rewires to another vertex y in the C-avoSI, the corresponding edge e in avoSI, which contains h as one of its two half-edges, breaks from x and reconnects to vertex y.
The process we construct on G has the same law as avoSI. Indeed, an infected half-edge can be rewired in C-avoSI if and only if it has not been paired. This exactly corresponds to the notion of 'unstable' that we used in the construction of avoSI process. Hence Lemma 3.1 follows.
From now on we will not distinguish between the C-avoSI and the avoSI since they are equivalent. The avoSI process stochastically dominates the evoSI process, as shown in the lemma below.

Lemma 3.2. The final size of infected vertices in avoSI stochastically dominates that in evoSI.
We couple the evoSI and avoSI as follows. The evoSI is constructed in the same manner as we did in the comparison of evoSI and delSI. See Section 2.1. We will use the variables T e,ℓ , R e,ℓ , U e,ℓ , ℓ ≥ 1 (defined at the beginning of Section 2.1) in the construction of avoSI to couple evoSI and avoSI.

Construction of avoSI.
For avoSI we need four sets to partition the set of all edges.
We use A to avoid confusion with the sets used in the construction of evoSI (not to be confused with the A t used in Section 2).
• Active edges with one end infected at time t, denoted by A a,1 t , are the edges at time t that connect an infected vertex and a susceptible vertex.
• Active edges with both ends infected, denoted by A a,2 t , are the unstable edges at time t that connect two infected vertices.
• Uninfected edges, denoted by A 0 t , connect two susceptible vertices.
• Inactive edges, denoted by A i t , consist of stable infected edges. Once an edge becomes inactive it remains inactive forever.
For each undirected edge e, we let random variables T e,ℓ , R e,ℓ , U e,ℓ be the same as those used in the construction of evoSI. We set S e,ℓ = min{T e,ℓ , R e,ℓ }. We also let V ′ e,ℓ be independent uniform random variables that take values in the two endpoints of e. To make it easier to describe the dynamics, suppose that at time τ a e,ℓ (the ℓ-th time e becomes active in avoSI) we have e = {x e,ℓ , y e,ℓ } with x e,ℓ infected and y e,ℓ susceptible.
The difference between evoSI and avoSI in terms of these clocks is as follows. For any edge e connecting x and y, once one endpoint x e,ℓ becomes infected, the clocks T e,ℓ and R e,ℓ start running. If the other endpoint y e,ℓ also becomes infected through other edges at (relative) time w e,ℓ < S e,ℓ (here 'relative' means we only count the time after the infection of x), then we replace the clocks T e,ℓ and R e,ℓ by T ′ e,ℓ = T e,ℓ − w e,ℓ 2 and R ′ e,ℓ = R e,ℓ − w e,ℓ 2 since the rates are now twice as fast.
To see that this construction give the correct dynamics of avoSI. We note that conditionally on T e,ℓ , R e,ℓ > w e,ℓ , (T e,ℓ − w e,ℓ )/2 and (R e,ℓ − w e,ℓ )/2 are independent exponential random variables with parameters 2λ and 2ρ, respectively. This corresponds to an unstable I − I pair where each of the two I ′ s attempt to send infection to the other I at rate λ and rewire from the other I at rate ρ. The variable V ′ e,ℓ corresponds to the vertex with small rewiring time.
If T e,ℓ > R e,ℓ then T ′ e,ℓ > R ′ e,ℓ and vice versa. Using this and the fact that the uniform variable U e,ℓ is the same in evoSI and avoSI we get the following proposition: If edge e is rewired in evoSI, then e must be rewired to the same vertex in avoSI (as long as either endpoint of e becomes infected in avoSI). Also, the time it take to e to be rewired in evoSI is not smaller than the time in avoSI.
We now formally construct the avoSI process by induction.
Initial step. At time 0, a randomly chosen vertex u 0 is infected. . There are no changes for other edges.
The avoSI process stops when there are no active edges.
Proof of Lemma 3.2. We now prove by induction that all vertices infected in evoSI are also infected in avoSI and actually they are infected earlier in avoSI than evoSI.
The induction hypothesis holds for the first vertex since initially u 0 is infected in both evoSI and avoSI. Suppose the induction holds up to the k-th infected vertex in evoSI. Assume at time t, y becomes the (k + 1)-th infected vertex in evoSI. We assume that y is infected by vertex x through edge e. Note that e has possibly gone through a series of rewirings before connecting vertex y. We assume that e connects vertices x ℓ and y ℓ after the (ℓ − 1)-th rewiring. We also assume when x infects y through e, e has been rewired r times. This implies that T e,ℓ > R e,ℓ for 1 ≤ ℓ ≤ r and T e,r+1 < R e,r+1 . We let m(x) = inf{i : x ∈ {x k , y k } for all i ≤ k ≤ r + 1}. We now divide the analysis into two cases: m = 1 and 1 < m ≤ r + 1. Case 1. m = 1 so that y = y k for all 1 ≤ k ≤ r + 1. If we assume that y has not been infected by time t, then x 1 , . . . x r must have been infected at the time that the rewiring occurred and x r+1 = x infected y at time t in evoSI. By the induction hypothesis we see that x 1 , . . . , x r+1 are also infected in avoSI. By Proposition 1 and (3.1.1), e breaks its connection with x 1 and reconnects to x 2 , then to x 3 and after r rewirings to x r+1 . If y is already infected before x r+1 sends an infection to it than we are done. Otherwise since T e,ℓ+1 < R e,ℓ+1 we see that x r+1 will send an infection to y in avoSI as well. In any case we have proved that x will also be infected in avoSI and is infected earlier. For a picture see Figure 6.
Case 2. If m > 1, then again by Proposition 1, the induction hypothesis and (3.1.1) we see that in the avoSI picture, e will be rewired at least r times and y becomes an endpoint of e exactly after m − 1 rewirings. After this point we can repeat the analysis in the case of m = 1 to deduce that x is also infected no later than t in avoSI. For a picture see Figure 7.   Figure 7: Illustration of the case m > 1. We assume that at the time of the rewiring x 1 , y 1 and x 2 are infected. Since one flips coins to determine the end that rewires, the sequence of of edges (x ′ k , y ′ k ) in avoSI is different from the edges in evoSI. However, thanks to the use of U e,ℓ to determine the new endpoint, the second rewiring brings the edge to y, and there is a correspondence between the vertices in the two processes indicated by the drawing.
We now show that avoSI and delSI actually have the same critical value (and thus also have the same critical value as evoSI by Theorem 1.6). Proof. As we will see, Lemma 3.3 follows by repeating the proof of Theorem 1.6. Since avoSI stochastically dominates evoSI (in terms of final epidemic size) and evoSI dominates delSI, avoSI must also dominate delSI. We claim that if we run avoSI on a tree and no edge is rewired to vertices that are infected up to time t, then there are no unstable I − I pairs up to time t. This immediately implies that the evolution of the avoSI process is equal to that of the evoSI process starting from the same initially infected vertex up to time t. To prove the claim, suppose there is an unstable I − I pair connecting vertex x and y, then there must be two infection paths that lead to the infections of x and y.
Here an infection path for x is just a sequence of vertices u 0 → u 1 → · · · → x where the former vertex in the chain infects the latter. Since the edge between x and y is unstable, we see that if we consider the union of the two infection paths together with the edge (x, y) then we get a cycle of infected vertices. This contradicts with the assumption that the original graph is a tree and no edges are rewired to vertices infected by t (so that rewirings will not help create any cycle of infected vertices).
As we mentioned in the proof of Lemma 2.2, by Lemma 2.12 in [23], whp the subgraph obtained by exploring the neighborhoods of n 3/8 / log n vertices of any fixed vertex is a tree. The proof of Theorem 1.6(i) implies that with high probability no edge is rewired to infected vertices up to step O(n 1/3 ) (i..e, up to the exploration of the neighborhoods of O(n 1/3 ) vertices). Hence the condition of the above claim is satisfied. Using the conclusion of the claim we see that whp we can couple the avoSI process and evoSI process such that they coincide up to step O(n 1/3 ). Therefore the proof of Theorem 1.6 also applies to the comparison of avoSI and delSI and hence Lemma 3.3 follows.

Tightness of {S t,k /n, t ≥ 0} n≥1
We first consider S t,k , the number of susceptible vertices with k half-edges at time t in the original avoSI process (i.e., without the time change). We have the following To explain the terms 1. At rate λ X I,t infections occur. The infected half-edge attaches to a susceptible vertex with k half-edges, which we call an S k , with probability k S k,t /( X t − 1). The −1 in the denominator is because the half-edge will not connect to itself.
2. At rate ρ X I,t rewirings occur. If k ≥ 1, the half-edge gets attached to an S k−1 with probability S k−1,t /n, promoting it to an S k . 3. If the rewired half-edge gets attached to an S k , which occurs with probability S k,t /n, it is promoted to an S k+1 and an S k is lost.
4. If Z t is a Markov chain with generator L then Dynkin's formula implies See Chapter 4, Proposition 1.7 in [8]. Fortunately, we do not need an explicit formula for the martingale. All that is important is that when f (Z t ) = S t,k , M ·,k has jumps equal to ±1.
The equation for S t,k can be then obtained by multiplying the first three terms in the right hand side of (3.2.1) by the time change, leading to Here M t,k is a time-changed version of the previous martingale so it is also a margingale with jumps ±1. Canceling common factors and dividing both sides of (3.2.2) by n we get  To do this we note that the expected value of quadratic variation of M t∧γn,k evaluated at time T , which is also equal to E(M 2 T ∧γn,k ), is bounded above by the expectation of total number of jumps in the whole avoSI process, which is equal to Here we have a factor of 2 in the denominator because each pairing event takes two halfedges and N j is the number of times that half-edge j gets transferred to another vertex.
Note that an infected half-edge gets rewired before being paired with probability at most ρ/(λ + ρ) and susceptible half-edge cannot get rewired unless the vertex it is attached to becomes infected. Thus, N j is stochastically dominated by a Geometric(ρ/(λ + ρ)) distributed random variable, so that for all j, E(N j ) ≤ C for some constant C. Therefore by L 2 maximal inequality applied to the submartingale M t∧γn,k , we obtain that where C is a constant whose value is unimportant. Since S 0,k /n ≤ 1, we know {S 0,k /n} n≥1 is a tight sequence of random variables. To establish tightness of {S t,k /n, t ≥ 0} n≥1 we need to show for any fixed ϵ, δ > 0, there is a θ > 0 and integer n 0 so that for n ≥ n 0 P sup |t1−t2|≤θ,t1,t2≤T S t1∧γn,k − S t2∧γn,k /n ≥ δ ≤ ϵ.
Using (3.2.7) and (3.2.5), we see that if we pick θ small and n large then the last line is ≤ ϵ. This proves (3.2.6) and thus completes the proof of tightness of the sequence {S t∧γn,k /n, t ≥ 0} n≥1 .

Convergence of {S t,k /n, t ≥ 0} n≥1
Note that the evolution for X t has the same transition rates as the time-changed SIR dynamics defined in [13] so their equation (3.4) also holds true in avoSI, which gives for In order to upgrade sup 0≤t≤T ∧γn to sup 0≤t≤γn , we note that for any ϵ > 0, we can pick a sufficiently large T so that m 1 exp(−2T ) < ϵ. Then by the monotonically decreasing property of X t , for n sufficiently large, which proves the first equation of (1.6.2). By the tightness of {S t∧γn,k /n, t ≥ 0} n≥1 , we see for any subsequence of S t,k /n we can extract a further subsequence that converges in distribution to a process s t,k with continuous sample path. By the Skorokhod representation theorem we can assume the convergence is actually in the almost sure sense and we can also assume that X t /n converges a.s. to m 1 exp(−2t). Having established tightness, a standard argument implies that we can show the convergence of S t,k /n by establishing that the limit s t,k is independent of the subsequence. First consider the case k = 0. The first two terms on the right-hand side of (3.2.3) are 0, so using (3.2.4) and first equation of (1.6.2) we see that any subsequential limit s t,0 has to satisfy the equation Since z → exp(−2z) is Lipschitz continuous this equation has a unique solution. Repeating this process for k ≥ 1 we see that any subsequential limit s t,k of S t,k /n satisfies the differential equation Define g t,k = exp (kt + (α/2)(1 − exp(−2t))) s t,k , Making the change of variable s = α(1 − exp(−t)) and letting h s,k = g t,k , we see that h s,0 is constant in s and h ′ s,k = h s,k−1 , k ≥ 1, from which we see h s,k is a polynomial of degree k in s and for all ℓ ≤ k the ℓ-th derivative of h s,k at s = 0 equals h 0,k−ℓ . From this we obtain that for all k, The initial conditions are g 0,k = h 0,k = s 0,k = p k = P(D = k). It follows that and hence using definitions of s, g, and h where w = w(t) = exp(−t).

Summing the s t,k
We pause to record the following fact which we will use later. From the explicit expression for s t,k in (3.3.8), dropping the factor exp(−α(1 − w 2 )/2) ≤ 1 and writing k = (k − ℓ) + ℓ, sup t≥0 k≥K The ℓ = 0 term in the first sum is bounded by k≥K kp k .
The remainder of the two sums is bounded by Interchanging the order of summation in the double sum and letting m = k − ℓ, the double sum in (3.4

.2) is bounded by
Combining our calculations leads to We can use this bound to show that ∞ k=0 S t,k /n converges to ∞ k=0 s t,k as well as ∞ k=0 kS t,k /n converges to ∞ k=0 ks t,k . Since the proofs are similar, we only prove the second result. We fix a large number K and observe that k≥K kS t,k satisfies the Here M t,K is a martingale that satisfies where Q ℓ is the number of half-edges that vertex ℓ has before it becomes infected. This follows from the observation that there are two sources for the jump of k≥K kS t,k : • A susceptible vertex ℓ with at least K half-edges gets infected. Then k≥K kS t,k drops by the number of half-edges of vertex ℓ, which is Q ℓ . Each vertex can contribute to this type of jumps at most once.
• A half-edge of an infected vertex gets transferred to a susceptible vertex of degree at least K − 1. Then k≥K kS t,k increases by either ≤ K (if the vertex gaining a half-edge had K − 1 half-edges before) or 1 (if the vertex gainning a half-edge had at least K half-edges before). Vertex ℓ can contribute at most Q ℓ times to jumps of size 1 and at most once to jumps of size K.
Initially vertex ℓ has D ℓ half-edges. As time grows the half-edges of other vertices might be transferred to vertex ℓ, the number of which is dominated by Conditioning on the value of W we have (3.4.10) Dividing both sides of (3.4.10) by n, taking the square and using (a where we have also used the Cauchy-Schwarz inequality to conclude that for any function If we use E to denote the conditional expectation with respect to the σ-algebra generated by X 0 , then for any ϵ > 0, using equation (3.4.9) we can find a constant (depending on ϵ) C 3.4.12 > 0 so that P E sup 0≤z≤t∧γn M 2 z,k > C 3.4.12 n ≤ ϵ. (3.4.12) We then take another constant C 3.4.13 such that P(X 0 /n > C 3.4.13 ) ≤ ϵ. Taking the conditional expectation of (3.4.11) with respect to X 0 we see on the event  Gronwall's inequality gives (3.4.15) provided that β(t) ≥ 0 and α(t) is nondecreasing. So applying (3.4.15) we have To control the first term on the right we use the convergence of S t,K−1 /n to s t,K−1 in probability as well as the bounded convergence theorem (since S t,K−1 /n ≤ 1) to obtain It follows that Using P(Ω 1 ) ≥ 1 − 3ϵ and the Chebyshev's inequality, we see that with probability ≥ 1 − 4ϵ, sup 0≤t≤T ∧γn k≥K kS t,k n ≤ ϵ.
Fixing t and ϵ and using the triangle inequality, we get  By first choosing K large enough and then n large enough we can make both the first and second terms on the right hand side of (3.4.18) smaller than ϵ with probability at least 1 − 4ϵ. The third term can also be made smaller than ϵ using (3.4

.3).
Since ϵ is arbitrary we see that To find ∞ k=0 s t,k and ∞ k=0 ks t,k , recall that we set w = w(t) = exp(−t) and G(w) = E(w D ). The limit of the fraction of susceptible nodes s t satisfies The limit of (scaled) number of susceptible half-edges satisfies . (3.4.20) Extension to time γ n . We have proved the second and third statements of (1.6.2) with 0 ≤ t ≤ γ n replaced by 0 ≤ t ≤ T ∧ γ n for any fixed T . To upgrade this to 0 ≤ t ≤ γ n , note that ∞ k=0 kS t,k ≤ X t . Picking a large T satisfying (m 1 + α) exp(−T ) ≤ ϵ and re-using equation (3.3.2) we obtain that P γ n > T, sup kS t,k /n > 2ϵ ≤ P(γ n > T, X T /n > 2ϵ) ≤ ϵ. is slightly more complicated. Again we fix a large T such that (m 1 + α) exp(−T ) ≤ ϵ and We first estimate the term S T,0 for large T . Using the weaker version (i.e., with sup 0≤t≤T ∧γn ) of the second equation of (1.6.2) we see that for n large enough, P γ n > T, S T,0 /n − exp −α/2 (exp(−T ) − 1) 2 G(exp(−T )) > ϵ ≤ ϵ. On the event {γ n > T }, sup T ≤t≤γn S t,0 − S T,0 can be bounded by X T , since in order to lose a susceptible vertex of degree 0 there must be a half-edge transferred to it. It follows that P γ n > T, sup T ≤t≤γn S t,0 − S T,0 /n > 2ϵ ≤ P γ n > T, X T /n > 2ϵ ≤ ϵ.
which proves the second equation of (1.6.2) and concludes the proof of (1.6.2).

Proof of Theorem 1.13
Recall for all t ≤ γ n we have X t ≥ X S,t and at γ n we have X t = X S,t since γ n is the time that we run out of infected half-edges and the dynamics stop. Note that by the We can rewrite f as Taking the derivative we have Theorem 1.6 tells us that in the supercritical case, we have λ > (ρm 1 )/(m 2 − 2m 1 ), and hence f ′ (1) < 0, which implies that f is positive on (1 − δ, 1) for some δ > 0. Theorem 1.6 also shows that when η > 0 is small, lim n→∞ P(I ∞ /n > η) = q(λ) > 0, (3.5.2) where q(λ) is the survival probability of the two-phase branching processZ m (defined in Section 1.2). Let t η < δ be some small number depending on η such that Conditionally on I ∞ /n > η for some small η, the second equation of (1.6.2) implies that with high probability γ n is also bounded from below by t η (depending on η), since otherwise we would have lim sup n→∞ P(γ n < t η , I γn /n > η) where the last equality is due to (3.5.3) and the fact that S γn + I γn = n. We have also used the definition of γ n so that I ∞ = I γn since no more vertices can be infected after γ n .
Recalling the definition of the σ in statement of Theorem 1.13 we see that for any ϵ > 0, inf tη<t<− log(σ+ϵ) x t x S,t > 1, which implies that for some ϵ ′ > 0 and all t η < t < − log(σ + ϵ), The first and third equations of (1.6.2) and (3.5.4) imply that where we have used the fact that X γn = X S,γn . Since we have already shown conditionally on I ∞ /n > η whp γ n > t η , we see that lim n→∞ P(γ n > − log(σ + ϵ)|I ∞ /n > η) = 1.

Proof of Theorem 1.14
From (3.5.1) we see that as λ → λ c we have f ′ (1) → 0. The second derivative of f is given by Since the terms with 1 − w vanish at w = 1, inserting the values of G(1), G ′ (1), G ′′ (1) and G ′′′ (1) we get From this, we see that at α c Using µ 1 = m 1 , µ 2 = m 2 − m 1 this can be written as (3.6.1) By Theorem 1.13 it suffices to prove that in the case ∆ < 0, σ converges to 1 as λ → λ c . Equation (3.6.1) and the assumption ∆ < 0 imply that for λ close to λ c , in a (non-shrinking) neighborhood of 1, f ′′ (w) has to be bounded from above by some negative constant. Since f ′ (1) converges to 0 as λ → λ c we conclude that for any fixed w < 1 and all λ sufficiently close to λ c one can find w ∈ (w, 1) so that f ( w) < 0. Using the definition of σ we see σ > w > w. Letting w → 1, we see that σ has to converge to 0 as λ → λ c and thus ν converges to 0. Hence we have a continuous phase transition.

AB-avoSI
Roughly speaking, the avoSI process serves as an upper bound because certain I − I pairs can rewire, which may leads to additional infections. To get a lower bound, we need to find a way to ensure rewired I − I edges will not transmit infections. This motivates the AB-avoSI process defined as follows. For each half-edge h we give it two indices: • The infection index A(h, t) = 0 if h has not been infected by time t. If i first becomes an infected half-edge at time s, then we set A(h, t) = s for all t ≥ s.
• The rewiring index B(h, t) = 0 if the half-edge h has not rewired by time t. If h gets rewired at time s, then we update the value of B(h, s) to be s, no matter whether h has been rewired before or not. In other words, if we let τ m (h) be the time when h is rewired for the m-th time (possibly ∞) with τ 0 (h) = 0, then B(h, t) = τ m (h) for τ m (h) ≤ t < τ m+1 (h).
We define the C-AB-avoSI process as follows. As in Section 3.1, C is for coupled.
• At rate λ each infected half-edge h 1 pairs with a randomly chosen half-edge. Suppose h 1 gets paired with half-edge h 2 at time t. If h 2 is susceptible and B(h 2 , t) < A(h 1 , t) then the vertex associated with half-edge h 2 becomes infected. Otherwise h 1 will not pass infection to the vertex associated with h 2 . The reader will see the reason for this condition in the proof of Lemma 4.1. Note that if vertex y associated with h 2 changes from state S to I then all half-edges attached to y become infected.
• Each infected half-edge gets removed from the vertex that it is attached to at rate ρ and immediately becomes re-attached to a randomly chosen vertex.
Similarly to the relation between C-avoSI and avoSI, one can also define the AB-avoSI such that the C-AB-avoSI has the same law as the AB-avoSI on the configuration model.
The construction of the graph G for AB-avoSI follows the same route of avoSI (see the proof of Lemma 3.1). Given the graph G, we view each (full) edge as being composed of two half-edges and assign the two indices A(·, t) and B(·, t) as defined above to every half-edge. The evolution of AB-avoSI is then similar to avoSI, except that each time when infected vertex x tries to infect ssuceptible vertex y through edge e, we will compare A(h 1 , t) and B(h 2 , t), where h 1 is the half-edge of e with one end at x and h 2 is the other half-edge of e with one end at y. If A(h 1 , t) > B(h 2 , t) then the infection will pass through.
Otherwise the infection will not pass through, which means x has made an attempt but y remains uninfected. No matter whether the infection passes through or not, we let this S − I edge be deemed stable and it will not get rewired later on. Also from this point on, x will never pass infection to y through e. As a comparison, S − I edges in avoSI are always unstable and are subject to potential rewiring. One can show that evoSI stochastically dominates AB-avoSI. We would like to show that after time t 1 no infections can pass through e to create additional infected vertices. Denote the half-edge attached to x by h 1 and the other half-edge by h 2 . Since x and y must be infected before time t 1 , we see that, according to the definition of the infection index, for all t ≥ t 1 , A(h 1 , t) < t 1 , A(h 2 , t) < t 1 . We next consider the rewiring index. Since the half-edge h 2 is rewired at time t 1 , using the definition of rewiring index we see that for all t ≥ t 1 , B(h 2 , t) ≥ t 1 . This implies that infections cannot pass from the vertex associated to h 1 to the vertex associated to h 2 . Now we consider the other direction, i.e., from h 2 to h 1 . Note that x has already been infected. There are two possible cases: • If h 1 stays with x forever then the vertex associated with h 1 is always infected after time t 1 (and hence there is no additional infected vertex). • If h 1 is rewired to some other vertex x ′ at time t 2 > t 1 then for all t > t 2 , which implies that after time t 2 infections cannot go from the vertex associated with h 2 to the vertex associated with h 1 .
In both cases there will be no additional infected vertices stemming from the rewiring of e. Thus we have completed the proof of Lemma 4.1.
We end this section by showing AB-avoSI dominates delSI. This is used in the proof Lemma 1.12.

Lemma 4.2.
There exists a coupling of AB-evoSI and delSI such that if a vertex is infected in AB-evoSI then it is also infected in delSI.
Proof. This can be proved in a similar way to the proof of Lemma 1.2. We construct AB-avoSI using the variables {T e,ℓ , R e,ℓ , V ′ e,ℓ , ℓ ≥ 1} as we did in the proof of Lemma 3.2 except that for AB-avoSI certain infections may not pass through depending on the relative size of infection index and rewiring index. We also construct delSI using T e,1 and R e,1 as we did in the proof of Lemma 1.2. We can then prove Lemma 4.2 by repeating the induction argument used in the proof of Lemma 1.2. Note that if edge e is initially present between x and y and T e,1 < R e,1 then either end of e can be infected by the other end because the rewiring indices for both half-edges of e are equal to 0 at the time of infection. Therefore we don't need to worry about infections not being transmitted successfully in AB-evoSI.

Moment bounds
LetX t andX I,t be the number of total half-edges and infected half-edges in the AB-avoSI process. As in the analysis of avoSI, we multiply the original transition rates by (X t − 1)/(λX I,t ). We will use a hat to denote the quantities after the time change. The evolution equation for X t has the same form as avoSI and hence the first equation of (1.6.2) also holds for AB-avoSI. Now we consider the evolution of the number of susceptible half-edges X S,t . We need a bit more notation to describe this. For half-edge i, we let I(i, t) = 1 if i is an infected half-edge at time t (which also means it hasn't been paired) and I(i, t) = 0 otherwise. We can define S(i, t) similarly. We also let S(i, k, t) = 1 if i is attached to a susceptible vertex with k half-edges at time t. Finally, we let v(j, t) be the vertex that half-edge j is attached to at time t and D(j, t) be the number of half-edges attached to v(j, t) at time t.
We first write down the equation for S t,k . To reduce the size of formulas we let Reasoning as in the derivation of (3.2.3) gives where M t,k is a martingale. Summing (4.2.2) over k from 0 to ∞ and noting that the second and third term cancel, we get Multiplying both sides of (4.2.2) by k and summing over k, we get for some martingale term M 2,t . Analogously, if we multiply both sides of (4.2.2) by k 2 , k 3 and k 4 , respectively, then we get The main result of this section is the following lemma.  Let Q x be the number of half-edges that vertex x originally has plus the half-edges that has been rewired to vertex x before x becomes infected. Let D x (t) be the number of half-edges that x has at time t. We necessarily have D x (t) ≤ Q x as long as x is susceptible at time t.
Note that the jumps of ∞ k=0 k 4 S t,k have the following three sources.
• An infected half-edge pairs with a susceptible half-edge attached to a vertex x with D x (t) half-edges and passes the infection to x. This decreases ∞ k=0 k 4 S t,k by D x (t) 4 . Such type of jumps can occur at most once for each susceptible vertex.
• An infected half-edge pairs with a susceptible half-edge attached to a vertex x with D x (t) half-edges but does not pass the infection. This decreases Such type of jumps can occur at most Q x times for vertex x.
• An infected half-edge is rewired to a susceptible vertex x with D x (t) half-edges.
This gives an increase of x to ∞ k=0 k 4 S t,k . Such type of jumps can happen at most Q x times for vertex x.
It follows from the above analysis that the quadratic variation of M 5,t is bounded by   x . (4.2.14) As we argued in the proof of (3.4.7), if V = Binomial( X 0 , 1/n), then Q x is dominated by D x + V . To bound the fifth moment of the sum we note that if Y and Z are nonnegative random variables, We claim that the 5-th moment of Binomial(m, p) is bounded by mp + · · · + (mp) 5 .
To see this, let Y 1 , . . . , Y m be i.i.d. Bernoulli variable with mean p.
p # of distinct elements among i1,...,i5 . (4.2.16) Note that the number of ordered tuples (i 1 , . . . , i 5 ) such that there are ℓ distinct elements among them is bounded by m ℓ . We conclude that (4.2.17) which verifies the claim. Using this claim we have that To prove equation (4.2.9), define the event Ω n to be ∞ k=0 k i S 0,k − n ∞ k=0 k i p k ≤ n, sup 0≤t≤γn∧1 |M i,t | ≤ n, for i = 1, 2, 3, 4 . The assumption that E(D 5 ) < ∞ and equation (4.2.8) imply that P(Ω n ) → 1 as n → ∞. By the definition of G i,j in (4.2.1), Here D r (t) is the number of half-edges that vertex r has at time t. Using (4.2.5) and On the event Ω n , we have that ∞ k=0 k 2 S 0,k ≤ (m 2 + 1)n, X S,u ≤ X 0 ≤ (m 1 + 1)n and M 3,t ≤ n. Therefore, using (4.2.23) we see that, there exsits a constant C 4.2.24 such that On the event Ω n , using (4.2.24), we see that, for all 0 ≤ t ≤ γ n ∧ 1, We now turn to the proof of equation (4.2.10). Set Note that H(0) = 0. Using Dynkin's formula, where M 6,t is a martingale associated with H(t) and h(t) is the rate of change of H(t). We now control h(t) and M 6,t by analyzing the jumps of H(t) (H(t) is a pure jump process).
Note that there are three types of jumps: • An infected half-edge pairs with a half-edge attached to susceptible vertex x and makes x infected. This does not increase H(t) and thus makes a non-positive contribution to h(t). The absolute value of the jump size of H(t) is bounded by D x (t) 3 where D x (t) is the number of half-edges that x has at time t. For each vertex x such jumps can happen at most once. • An infected half-edge pairs with a half-edge attached to susceptible vertex x but x stays susceptible after the pairing. This does not increase H(t) and thus makes a non-positive contribution to h(t). The absolute value of the jump size of H(t) is bounded above by To see this, note that the loss of a half-edge j attached to x makes a twofold contribution to H(t). First, j is no longer a half-edge so H(t) has to decrease by (D(j, t) − 1) 2 = (D x (t) − 1) 2 . Second, the for each of the remaining D x (t) − 1 half-edges attached to x, its contribution to For each vertex x such type of jumps can occur at most Q x times. • An infected half-edge is rewired to a susceptible vertex x. This increases H(t) by at most The rate that x receives a rewired half-edge is equal to X 0 λ X I,t ρ X I,t 1 n = ρ X 0 λn ≤ (m 1 + 1)ρ λ on the event Ω n (defined in (4.2.21)). Here the factor of 1/n comes from Poisson thinning since each half-edge is a rewired to a uniformly chosen vertex independently. For each vertex x such type of jumps can occur at most Q x times.
Therefore on Ω n we have Equation (4.2.9) shows that with high probability ∞ k=0 (k + 1) 4 S t,k ≤ C 4.2.9 n. Since Ω n also holds with high probability, we deduce that lim n→∞ P t 0 h(s)ds ≤ C 4.2.28 nt, ∀0 ≤ t ≤ γ n = 1. (4.2.28) The above analysis of the jumps of H(t) also implies that the quadratic variation of M 6,t is bounded by We now bound the 2/3-th moment of the quadratic variation. Applying (4.2.13) with p = 3/2 and a i = (Q i + 1) 6 gives that Using this and the Burkholder-Davis-Gundy inequality ( [15,Theorem 7.34] with p = 4/3),
The proof of (4.3.3) is parallel to the proof of (4.3.2), except that we now replace the second line of (4.3.12) by E(u) (defined in (4.3.1)). We can do this because which is true by the definition of E(t) in (4.3.1).
which follows from the fact that D(j, t) ≥ 1, we see that   The definition of L(t) and the fact D(j, t) ≤ D(j, t) 2 imply that Using the fact D(j, t) ≤ D(j, t) 2 again, (4.2.10) implies that lim n→∞ P(L(t) ≤ n(C 4.2.10 t + ϵ), ∀0 ≤ t ≤ γ n ∧ 1) = 1.  The bound provided in (4.3.25) is not enough for our purpose (though we will also use it in the proof of Theorem 1.15). We will prove refined bounded in the next section.

More refined bounds
Equation (4.3.3) implies that we can get a lower bound for X I,t if we can upper bound the term t 0 E(u)du. To this end, we let b be some number in (0, 1) to be determined. We can decompose E(t) into two parts: Recalling the definition of L(t) in (4.3.22), we see that In the next two lemmas we give bounds on L(b, t) and X(I, b, t).  The first term in the second line of (4.4.5) has already been controlled by equation (4.2.10), i.e., Let N (t) be the number of rewiring events that occur by time t. Then we have  Now we write down the evolution equation for N (t) and M 7,t is some martingale.
We claim that {A(i, t) ≤ bt, I(i, t) = 1} ⊂ ∪ 4 k=1 H k (i, t). Indeed, either I(i, bt) = 1 or S(i, bt) = 1 must hold. The case of S(i, bt) = 1 corresponds to H 4 (i, t). On the other hand, if I(i, bt) = 1 and I(i, t) = 1, then there are three possible cases: i didn't get rewired in [bt, t], i was rewired to an infected vertex or i was rewired to a susceptible vertex which later became infected. The first case case corresponds to H 1 (i, t) while the second and third case are covered in H 2 (i, t) and H 3 (i, t), respectively.
It remains to control H 3 (t) and H 4 (t). For any vertex x, let R(x) be the indicator function of the event that vertex x has received at least one rewired edge when x first becomes infected and let Q x be the number of half-edges that x has just before it becomes infected. Let R(x, t) be the indicator of the event that x has received at least one rewired half-edge by time t. Then we have, by the definitions of H 3 (t) and H 4 (t),   Let D x (u) be the number of half-edges of vertex x at time u and D(j, u) the number of half-edges that v(j, u) has at time u (recall that v(j, u) is the vertex that half-edge j is attached to at time u). The process N (bt, u), bt ≤ u ≤ t has a positive jump whenever a susceptible vertex with at least one rewired half-edge gets infected. The probability that x is infected (given an infection event occurs) is equal to D x (u)/( X u − 1) and the contribution to N (bt, t) is equal to R(x, u)D x (u). Thush t (u) satisfies h t (u) ≤ λ X I,u X u − 1 λ X I,u  In the first step of (4.4.43) we used the definition of R(x, u) so that  The definition of N (bt, t) as the right hand side of (4.4.40) implies that we can upper bound the quadratic variation of M 8,t by n i=1 Q 2 i where Q i is the number of half-edges that vertex i has before it becomes infected. Using this and the Burkholder-Davis-Gundy      To make the computation easier to write we note that when t ≤ t 0 ≤ 1,