A Generalized SIS Epidemic Model on Temporal Networks with Asymptomatic Carriers and Comments on Decay Ratio
AA Generalized SIS Epidemic Model on Temporal Networks withAsymptomatic Carriers and Comments on Decay Ratio
Ashish R. Hota and Kavish Gupta
Abstract — We study the class of SIS epidemics on temporalnetworks and propose a new activity-driven and adaptiveepidemic model that captures the impact of asymptomatic andinfectious individuals in the network. In the proposed model,referred to as the A-SIYS epidemic, each node can be inthree possible states: susceptible, infected without symptomsor asymptomatic and infected with symptoms or symptomatic.Both asymptomatic and symptomatic individuals are infectious.We show that the proposed A-SIYS epidemic captures severalwell-established epidemic models as special cases and obtainsufficient conditions under which the disease gets eradicatedby resorting to mean-field approximations.In addition, we highlight a potential inaccuracy in thederivation of the upper bound on the decay ratio in the activity-driven adaptive SIS (A-SIS) model in [1] and present a moregeneral version of their result. We numerically illustrate theevolution of the fraction of infected nodes in the A-SIS epidemicmodel and show that the bound in [1] often fails to capture thebehavior of the epidemic in contrast with our results.
I. I
NTRODUCTION
The susceptible-infected-susceptible (SIS) epidemic is oneof the most well-studied class of spreading processes onnetworks [2], [3]. Early work on SIS epidemics focusedon analyzing both deterministic [4] and stochastic [3], [5]dynamic evolution of the epidemic states; often resorting tomean-field approximations for analytical tractability. Mostof the existing work has analyzed the epidemic dynamicson static networks; both deterministic as well as large-scalecomplex networks with a structured population [3], [5].However, the contact pattern in the human population isdynamic and time-varying. Furthermore, during the preva-lence of an infectious disease, individuals often take precau-tions and reduce their social activities to protect themselvesand others from becoming infected. Thus, the characteristicsof the network or contact pattern evolves in a time-scale thatis comparable to the evolution of the epidemic. Consequently,several recent works have analyzed epidemic processes ontemporal or dynamical networks [6]–[11].In this work, we consider the class of SIS epidemics withinthe activity-driven network paradigm which is a relativelysimple yet expressive paradigm for analyzing the evolutionof epidemics and contact pattern in a comparable time-scale[12], [13]. Our work is motivated by and builds upon therecent works [1], [14] that study SIS epidemics and theirclose variants on activity-driven networks. Specifically, [1]defines the discrete-time activity-driven adaptive-SIS model (activity-driven A-SIS model), derives an analytical upper
The authors are with the Department of Electrical Engineering, Indian In-stitute of Technology (IIT) Kharagpur, India. E-mail: [email protected],[email protected]. bound on the decay ratio of the infection probabilities ofthe nodes and proposes tractable optimization problems foroptimal containment of the epidemic by minimizing thebound on the decay ratio. Similarly, in [14], the authorsstudy a continuous-time SAIS epidemic with an additionalstate that captures individuals who are alert and protectthemselves from the epidemic. The authors derive conditionsfor epidemic persistence and investigate optimal policies tomitigate the epidemic by reducing activation probabilities ofinfected nodes and prompting self-protective behavior.Our work is motivated by infectious diseases where asubset of infected individuals do not develop symptoms de-spite being infectious, i.e., they act as asymptomatic carriers;examples include COVID-19 [15], [16] and Ebola [17]. Suchindividuals are often not aware of being infected and do notreduce their activity and contact patterns. As a result, suchdiseases are often challenging to contain. However, the abovecharacteristic is not captured by the classical SIS epidemicmodel and its well-established variants that have additionalstates such as alert, exposed, etc. While recent papers [18],[19] have highlighted the impacts of such asymptomaticcarriers on the evolution and control of epidemics, rigorousand quantitative analysis of the above characteristic are fewin the existing work on epidemics (on temporal networks).In this paper, we propose a new activity-driven andadaptive generalized SIS epidemic model, referred to asthe A-SIYS epidemic, where we treat asymptomatic andsymptomatic individuals as distinct infection states (seeSection II for a formal definition and discussion). Our modelcaptures several well-established epidemic models as specialcases. Furthermore, in our setting, each node potentiallychooses a different number of nodes to connect to; this isin contrast with the homogeneity assumption in [1], [14].We derive a linearized dynamics that upper bounds theMarkovian evolution of the epidemic states via a mean-field approximation and present sufficient conditions underwhich the epidemic gets eradicated. Our results and prooftechniques are inspired by the analysis in [1].As a second contribution, we highlight a potential inaccu-racy in the derivation of the upper bound on the decay ratioof the A-SIS epidemic model in [1] and obtain a counterpartof their result for a more general setting where nodes choosedifferent numbers of other nodes to connect to (Section III).We then simulate the epidemic models for various parametersettings and show that the bound obtained in [1] does notalways capture the behavior of the epidemic, in contrast withour results (Section IV). We conclude with a discussion onopen problems and avenues for future research (Section V). a r X i v : . [ phy s i c s . s o c - ph ] J u l XY β x ( X ) , β y ( Y ) δ x νδ y Fig. 1:
Probabilistic evolution of states in the A-SIYS epidemic model.Self-loops are omitted for better clarity. See Definition 1 for the formaldefinition. Red indicates that both X and Y are infected states. II. A
CTIVITY -D RIVEN A DAPTIVE
SIYS E
PIDEMIC
In this section, we formally define the activity-drivenadaptive SIYS (A-SIYS) epidemic model. Let V = { v , v , . . . , v n } denote the set of n nodes. Each noderemains in one of the three possible states: susceptible ( S ),infected without symptoms or asymptomatic ( X ) and infectedwith symptoms or symptomatic ( Y ). Both asymptomatic andsymptomatic individuals are infectious, which captures thecharacteristics of certain epidemics such as COVID-19.The states evolve in discrete-time. If at time t ∈ { , , . . . } ,node v i is susceptible (respectively, asymptomatic and symp-tomatic), we denote this by v i ( t ) ∈ S (respectively, v i ( t ) ∈ X and v i ( t ) ∈ Y ). Given a network or contact pattern, theprobabilistic state evolution is defined below. Definition 1:
Let β x , β y , δ x , δ y , ν ∈ [0 , be constantspertaining to infection, recovery and transition rates. Thestate of each node v i evolves as follows.1) If v i ( t ) ∈ S , then v i ( t + 1) ∈ X with probability β x foreach asymptomatic neighbor and with probability β y for each symptomatic neighbor independently of otherneighbors.2) If v i ( t ) ∈ X , then v i ( t + 1) ∈ S with probability δ x and v i ( t + 1) ∈ Y with probability ν (1 − δ x ) .3) If v i ( t ) ∈ Y , then v i ( t + 1) ∈ S with probability δ y .The state remains unchanged otherwise. (cid:3) The possible transitions of the states are illustrated inFigure 1. Thus, in our model, both asymptomatic and symp-tomatic nodes can potentially infect a susceptible node, albeitwith different probabilities ( β x and β y , respectively). Uponbeing infected, a susceptible node becomes asymptomatic.From there on, it can either get cured and become suscep-tible with probability δ x , and if not, it transitions to thesymptomatic state with probability ν . Thus ν − captures thedelay in onset of symptoms. The curing rate for symptomaticnodes is δ y . Thus, in our model, a node can get infected andcured without ever exhibiting symptoms.With the above definition in place, we now formallydefine the activity-driven and state-dependent evolution ofthe network or contact pattern and the epidemic states ofindividual nodes. As discussed above, our model builds uponthe formulation in [1] for the A-SIS epidemic model. Definition 2:
For each node v i ∈ V , let a i , χ i , π i ∈ (0 , be constants referred to as the activity rate, adaptation factorand acceptance rate of v i , respectively. Let m i ≥ be thenumber of nodes v i attempts to connect to upon activation.Let β x , β y , δ x , δ y , ν ∈ [0 , be constants pertaining toinfection and recovery rates. The A-SIYS model is definedby the following procedures:1) At the initial time t = 0 , each node is in one of thethree possible states.2) At each time t = 1 , , . . . , each node v i randomlybecomes activated independently of other nodes withthe following probability: P ( v i becomes activated ) = (cid:40) a i , if v i ( t ) ∈ S ∪ X ,χ i a i , if v i ( t ) ∈ Y .
3) Node v i , upon activation, randomly and uniformlychooses m i other nodes independently of other acti-vated nodes. If v j is chosen by v i , an edge ( v i , v j ) iscreated with the following probability: P ( edge ( v i , v j ) is created ) = (cid:40) , if v j ( t ) ∈ S ∪ X ,π j , if v j ( t ) ∈ Y . These edges are discarded at time t + 1 .4) Once the edges are formed following the above pro-cedure, the states of the nodes get updated followingDefinition 1.5) Steps 2-4 are repeated for each time t ≥ . (cid:3) Thus, for node activation and link formation, susceptibleand asymptomatic (who are not aware of being infected)nodes behave in an identical manner. When χ i ∈ (0 , ,the probability of node v i getting activated when it issymptomatic is smaller than its activation probability whenit is susceptible or asymptomatic. This is potentially dueto sickness or reduction of activities by v i so as to notinfect others when it learns that it is infected. Similarly, asymptomatic node is less likely to accept an edge comparedto a susceptible or asymptomatic node. Remark 1:
The A-SIYS epidemic defined above is quitegeneral and captures the following models as special cases.1) ν = 0 , v i (0) / ∈ Y , ∀ v i ∈ V : In this case, the nodesnever enter the symptomatic state, and the activationand acceptance rates no longer state-dependent; thelatter parameters are a i and , respectively. Thus, theepidemic behaves as the classical SIS epidemic on anactivity-driven network (but non-adaptive).2) β x = 0 , δ x = 0 : Here, an asymptomatic node is notinfectious and eventually becomes symptomatic beforebecoming susceptible. In this regime, our model isthe activity-driven and adaptive analogue of the SEISepidemic [2] with X being the “exposed” state. (cid:3) A. State Evolution and Mean-Field Approximation
In order to analyze the evolution of the states inthe A-SIYS epidemic model, we define random variables S i ( t ) , X i ( t ) and Y i ( t ) associated with node v i that takevalues in the set { , } . Specifically, we define S i ( t ) = 1 if v i ( t ) ∈ S , X i ( t ) = 1 if v i ( t ) ∈ X and Y i ( t ) = 1 if i ( t ) ∈ Y . Since a node can only be in one of three possiblestates, we have S i ( t ) + X i ( t ) + Y i ( t ) = 1 . Similarly, wedefine a { , } -valued random variable A ij ( t ) which takesvalue if the edge ( v i , v j ) exists at time t . We also denoteby N x a Bernoulli random variable that takes value withprobability x ∈ [0 , . The state transition of v i under theA-SIYS epidemic model can now be formally stated as: S i ( t + 1) = S i ( t ) Π j (cid:54) = i (cid:2) − A ij ( t )( X j ( t ) N β x + Y j ( t ) N β y ) (cid:3) + N δ x X i ( t ) + N δ y Y i ( t ) , (1a) X i ( t + 1) = (1 − N δ x )(1 − N ν ) X i ( t )+ S i ( t ) (cid:2) − Π j (cid:54) = i (cid:2) − A ij ( t )( X j ( t ) N β x + Y j ( t ) N β y ) (cid:3) (cid:3) , (1b) Y i ( t + 1) = (1 − N δ x ) N ν X i ( t ) + (1 − N δ y ) Y i ( t ) . (1c)It is easy to see that S i ( t + 1) + X i ( t + 1) + Y i ( t + 1) = S i ( t ) + X i ( t ) + Y i ( t ) = 1 . We denote the probability ofnode v i being susceptible at time t by s i ( t ) , i.e., s i ( t ) := P ( v i ( t ) ∈ S ) = E [ S i ( t )] . The quantities x i ( t ) and y i ( t ) aredefined in an analogous manner.Note that the infection states follow a Markov processwith a n × n transition probability matrix with the state S i ( t ) = 1 for all v i ∈ V (i.e., the disease-free state) being theonly absorbing state. While analyzing the behavior of thismodel is computationally intractable, we rely on a mean-field approximation and upper bound the evolution of theinfection probability (both asymptomatic and symptomatic)via a linear dynamics. We then derive sufficient conditionsunder which the epidemic decays to the disease-free state.We start with the following result. Theorem 1:
Consider the activity-driven adaptive SIYS(A-SIYS) epidemic model defined in Definition 2. Let ¯ m i = m i / ( n − , and for all i, j , define the constants β ijx := β x [1 − (1 − a i ¯ m i )(1 − a j ¯ m j )] , (2) β ijy := β y [1 − (1 − a i ¯ m i π j )(1 − χ j a j ¯ m j )] . (3)Then, x i ( t + 1) ≤ δ cx (1 − ν ) x i ( t )+ (cid:88) j (cid:54) = i (cid:2) β ijx x j ( t )+ β ijy y j ( t ) (cid:3) , (4a) y i ( t + 1) = δ cx νx i ( t ) + (1 − δ y ) y i ( t ) , (4b)for all nodes v i and t ≥ with δ cx = 1 − δ x . (cid:3) Proof:
We compute expectation on both sides of (1b)and (1c) and obtain x i ( t + 1) = (1 − δ x )(1 − ν ) x i ( t )+ E (cid:104) S i ( t ) (cid:2) − Π j (cid:54) = i (cid:2) − A ij ( t )( X j ( t ) N β x + Y j ( t ) N β y ) (cid:3) (cid:3)(cid:105) , (5a) y i ( t + 1) = (1 − δ x ) νx i ( t ) + (1 − δ y ) y i ( t ) , (5b)and s i ( t ) = 1 − x i ( t ) − y i ( t ) . For the product term in theR.H.S. of (5a), the Weierstrass product inequality yields − Π j (cid:54) = i (cid:2) − A ij ( t )( X j ( t ) N β x + Y j ( t ) N β y ) (cid:3) ≤ (cid:88) j (cid:54) = i A ij ( t )( X j ( t ) N β x + Y j ( t ) N β y ) . Consequently, we have E (cid:104) S i ( t ) (cid:2) − Π j (cid:54) = i (cid:2) − A ij ( t )( X j ( t ) N β x + Y j ( t ) N β y ) (cid:3) (cid:3)(cid:105) ≤ (cid:88) j (cid:54) = i β x E [ A ij ( t ) S i ( t ) X j ( t )]+ β y E [ A ij ( t ) S i ( t ) Y j ( t )] . (6)We now focus on evaluating the expectation terms in theabove equation. Recall that A ij ( t ) is a random variable thatindicates the presence of the edge ( v i , v j ) at time t andis governed by the states of nodes v i and v j according toDefinition 2. In order to bound the expectation terms, weintroduce the following notation for events of interest: SX tij = “ v i ( t ) ∈ S and v j ( t ) ∈ X , ” (7) SY tij = “ v i ( t ) ∈ S and v j ( t ) ∈ Y , ” (8) Γ ti → j = “ v i is activated, chooses v j as neighbor at t.” (9)With the above notation in place, we have E [ A ij ( t ) S i ( t ) X j ( t )] = P ( A ij ( t ) = 1 | SX tij ) P ( SX tij ) , (10a) E [ A ij ( t ) S i ( t ) Y j ( t )] = P ( A ij ( t ) = 1 | SY tij ) P ( SY tij ) . (10b)We now focus on the first equation above and note that P ( A ij ( t ) = 1 | SX tij ) = P (Γ ti → j | SX tij ) + P (Γ tj → i | SX tij ) − P (Γ ti → j | SX tij ) P (Γ tj → i | SX tij )= 1 − [1 − P (Γ ti → j | SX tij )][1 − P (Γ tj → i | SX tij )] . Since the event SX tij states that v i is susceptible and v j isinfected without symptoms, according to Definition 2, theadaptation and acceptance of node v j is same as the casewhen it is susceptible. Therefore, we have P (Γ ti → j | SX tij ) = a i ¯ m i , P (Γ tj → i | SX tij ) = a j ¯ m j . Similarly for events conditioned on SY tij , we have P ( A ij ( t ) = 1 | SY tij ) = 1 − [1 − P (Γ ti → j | SY tij )] × [1 − P (Γ tj → i | SY tij )] . The event SY tij corresponds to v i being susceptible and v j being infected with symptoms. Therefore, the adaptationand acceptance of v j depend on the parameters χ j and π j ,respectively. Therefore, following Definition 2, we have P (Γ ti → j | SY tij ) = a i ¯ m i π j , P (Γ tj → i | SY tij ) = χ j a j ¯ m j . Finally, we note that P ( SX tij ) ≤ x j ( t ) , P ( SY tij ) ≤ y j ( t ) , since the event v j ( t ) ∈ X (respectively, v j ( t ) ∈ Y ) subsumesthe event SX tij (respectively, SY tij ).Substituting the above bounds and the expressions for theconditional probabilities obtained in (10), we obtain E [ A ij ( t ) S i ( t ) X j ( t )] ≤ [1 − (1 − a i ¯ m i )(1 − a j ¯ m j )] x j ( t ) , E [ A ij ( t ) S i ( t ) Y j ( t )] ≤ [1 − (1 − a i ¯ m i π j )(1 − χ j a j ¯ m j )] y j ( t ) . The result now follows upon substituting the above expres-sions in (6) and the definition of β ijx and β ijy .The above result shows that the evolution of the probabil-ity of a node being asymptomatic and symptomatic is upperbounded by a linear dynamics as stated in (4). The linearizeddynamics can be stated in a compact manner as follows.Let z ( t ) := [ x ( t ) (cid:62) y ( t ) (cid:62) ] (cid:62) ∈ [0 , n be the vector ofrobabilities corresponding to the infected states. From theabove theorem, we have z ( t + 1) ≤ (cid:20) A xx A xy A yx A yy (cid:21) z ( t ) =: A z ( t ) , (11)where each sub-matrix has dimension n × n . Specifically, A xx has diagonal entries (1 − δ x )(1 − ν ) and ( i, j ) -th entry as β ijx for j (cid:54) = i , A xy has diagonal entries and β ijy as the ( i, j ) -th entry with j (cid:54) = i , A yx := diag ((1 − δ x ) ν ) , and A yy := diag (1 − δ y ) . Consequently, we obtain a sufficient condition,stated below, under which the disease is eradicated. Theorem 2:
The decay ratio of the epidemic α := inf { γ : there exists C > such that || z ( t ) || ≤ Cγ t for all t ≥ and z (0) } is upper bounded as α ≤ ρ ( A ) where ρ ( A ) is the spectralradius of A . In particular, if ρ ( A ) < , lim t →∞ || z ( t ) || = 0 . (cid:3) The proof follows from the above discussion and standardarguments and is omitted in the interest of space. Note furtherthat A is a non-negative irreducible (since each node canpotentially choose any other node to connect to) matrix.Thus, ρ ( A ) corresponds to its largest eigenvalue which is realand positive following the Perron-Forbenius theorem [20].We now state the following corollaries of the above resultsthat correspond to certain special cases of our model. Corollary 1:
Suppose ν = 0 and v i (0) / ∈ Y , ∀ v i ∈ V .Then, y i ( t ) = 0 for all t ≥ and x i ( t ) ≤ (1 − δ x ) x i ( t ) + (cid:88) j (cid:54) = i β ijx x j ( t ) . Furthermore, α ≤ ρ ( A xx ) . (cid:3) The above setting corresponds to the classical SIS epi-demic on an activity-driven network discussed in Remark1. Note from the definition of β ijx that the matrix A xx consists of a diagonal matrix and a matrix of rank , andconsequently, its spectral radius can be explicitly derived.When symptomatic individuals completely stop interactingwith others, we have the following corollary. Corollary 2:
Suppose χ i = 0 and π i = 0 for all the nodes.Then, β ijy = 0 and we have z ( t + 1) ≤ (cid:20) A xx n × n A yx A yy (cid:21) z ( t ) , where n × n has all entries equal to . The bound on thedecay ratio is given by α ≤ max(1 − δ y , ρ ( A xx )) . (cid:3) The above regime corresponds to situations where symp-tomatic individuals are kept in strict isolation. Our analysisshows that even when the decay ratio pertaining to interac-tion among nodes (corresponding to ρ ( A xx ) ) is small, theepidemic eradication rate (dominated by − δ y ) can be slowif the recovery rate δ y is sufficiently small.As discussed earlier, the proposed model and the aboveanalysis is motivated by and builds upon the Activity-DrivenAdaptive SIS (A-SIS) epidemic proposed in [1]. In thefollowing section, we highlight a potential inaccuracy in thederivation of the upper bound on the decay ratio in [1]. III. A CTIVITY -D RIVEN
A-SIS E
PIDEMIC M ODEL AND P OTENTIAL I NACCURACY IN THE A NALYSIS OF [1]The activity-driven A-SIS epidemic is not a special case ofthe A-SIYS epidemic studied above, but is closely related.In the A-SIS epidemic defined in [1], a node v i is eithersusceptible or infected. A susceptible node becomes infectedwith probability β ∈ [0 , when it comes in contact with aninfected node (independently of other infected nodes) andan infected node recovers with probability δ ∈ [0 , . Thus,the state transition is a special case of Definition 1 when ν = 0 , δ x = δ and β x = β and X denoting the infected state.The models differ in the activity-driven adaptive networkformation process. While the A-SIYS epidemic distinguishesbetween asymptomatic and symptomatic infections, the A-SIS epidemic does not. Specifically, upon infection, node v i adjusts its activation and acceptance probabilities with thefactors χ i and π i , respectively as shown in points 2 and 3 inDefinition 2. Furthermore, each node upon activation chooses m other nodes to connect to, i.e., m i = m for all v i ∈ V .The rest of the steps are identical to those in Definition 2.We follow the terminology in [1] and model the state ofnode v i at time t as a random variable x i ( t ) := (cid:40) if v i is susceptible at time t, if v i is infected at time t. (12)Similarly, we define p i ( t ) := P ( v i is infected at time t ) andthe vector of infection probabilities for all nodes as p ( t ) . Theauthors in [1] define decay ratio as follows. Definition 3 (Definition 3.1 [1]):
We define the decay ra-tio of the activity-driven A-SIS model by α = inf { γ : there exists C > such that || p ( t ) || ≤ Cγ t for all t ≥ and x (0) } . The quantity α captures the persistence of infection amongthe nodes. In the A-SIS model, the states follow a Markovprocess and the actual decay ratio is the spectral radius ofthe n × n transition probability matrix. The authors in [1]upper bound the evolution of p i ( t ) by a linear dynamics andobtain an explicit upper bound on the decay ratio by notingthat it is the spectral radius of a matrix of rank .However, we believe that the derivation of the linearizeddynamics in Proposition 3.2 in [1] is inaccurate. We start ourdiscussion by first stating Proposition 3.2 from [1]. Proposition 1 (Proposition 3.2 [1]):
Let ¯ m = m/ ( n − , δ c = 1 − δ , and for all i , define the constants φ i = ¯ mχ i a i , ψ i = ¯ mπ i a i . Then, p i ( t + 1) ≤ δ c p i ( t ) + β n (cid:88) j =1 [1 − (1 − ψ i )(1 − φ j )] p j ( t ) (13)for all nodes v i and t ≥ . (cid:3) Potential inaccuracy in Proposition 3.2 [1]:
The proof of Proposition 3.2 in [1] follows largely analogoussteps as the proof of Theorem 1 above; the main distinctionbeing the absence of terms related to Y j ( t ) in [1]. We believehat the evaluation of P (Γ ti → j | Ξ ti,j ) in equation (3.19) in theproof in [1] is inaccurate. Note that Ξ ti,j := “ v i is susceptible and v j is infected at time t ” , in equation (3.16) in [1] and Γ ti → j is as defined in (9) above.Thus, P (Γ ti → j | Ξ ti,j ) is the probability that the edge ( v i , v j ) will be formed when initiated by the activated node v i when v i is susceptible and v j is infected. Thus, P (Γ ti → j | Ξ ti,j ) is the product of P ( v i is activated while it is susceptible ) and P (( v i , v j ) are neighbors when v j is infected ) . Follow-ing the definition of the A-SIS epidemic, we have P (Γ ti → j | Ξ ti,j ) = a i · ¯ mπ j (cid:54) = ψ i as ψ i = ¯ ma i π i . Since v j is infected at time due to theconditioning event Ξ ti,j , the probability of such an edge beingformed is ¯ mπ j not ¯ mπ i as considered in [1]. In other words,the probability that the edge ( v i , v j ) will be formed wheninitiated by the activated node v i depends on the acceptancerate of the node v j . Strengthening equation (3.21) in [1] when i = j : The authors claim that the inequality E [(1 − x i ( t )) A ij ( t ) x j ( t )] ≤ [1 − (1 − ψ i )(1 − φ j )] p j ( t ) , trivially holds when i = j . In fact, the event Ξ ti,i = “ v i is susceptible and v i is infected at time t ” is empty andas a result P (Ξ ti,i ) = 0 . Therefore, the bound can bestrengthened by treating E [(1 − x i ( t )) A ii ( t ) x i ( t )] = 0 . Implications:
The above potential inaccuracy has significant implicationon the bound derived in Theorem 3.3 in [1]. Specifically, theauthors build upon Proposition 3.2 and show that the vectorof infection probabilities evolves as p ( t + 1) ≤ (cid:2) (1 − δ ) I n + β [ n (cid:62) n − ( n − ψ )( n − φ ) (cid:62) ] (cid:3) p ( t )=: F p ( t ) , where I n is the identity matrix and n is the vector ofdimension n with all entries being . The authors then arguethat the spectral radius of F , denoted ρ ( F ) , is an upperbound on the decay ratio with ρ ( F ) = 1 − δ + βρ ( n (cid:62) n − ( n − ψ )( n − φ ) (cid:62) ) . Since n (cid:62) n − ( n − ψ )( n − φ ) (cid:62) is a matrix of rank , theauthors could obtain an explicit expression on its spectralradius and consequently on the bound on the decay ratio.However, due to the potential inaccuracy highlighted above,the linear dynamics that bounds the evolution of the vectorof infection probabilities is not necessarily a lower-rankedmatrix. Furthermore, the contributions related to optimalresource allocation for containing the epidemic rely on thebound on the decay ratio and may no longer be applicable.We now state the following theorem that addresses theabove inaccuracy and generalizes the result in [1]. Theorem 3:
Consider a generalization of the activity-driven A-SIS model where node v i , upon activation, choosesuniformly and randomly m i other nodes to connect to. Let ¯ m i = m i / ( n − , and for all i, j , define the constants φ i = χ i a i ¯ m i , ψ ij = a i ¯ m i π j . Then, p i ( t +1) ≤ (1 − δ ) p i ( t )+ β (cid:88) j (cid:54) = i [1 − (1 − ψ ij )(1 − φ j )] p j ( t ) (14)for all nodes v i and t ≥ . Furthermore, the decay ratio isupper bounded as α ≤ ρ ∗ := ρ ( F ∗ ) or the spectral radius ofthe matrix F ∗ with entries F ∗ ij = (cid:40) − δ, if i = j,β (cid:80) j (cid:54) = i [1 − (1 − ψ ij )(1 − φ j )] , if i (cid:54) = j. (15)In particular, if m i = m for every node v i , then theresult holds with φ i = χ i a i ¯ m and ψ ij = a i ¯ mπ j where ¯ m = m/ ( n − . (cid:3) The proof largely mirrors the proof of Theorem 1 and theanalysis in [1] with the above discussed aspects incorporated.We omit it in the interest of space.
Remark 2:
The upper bound on the decay ratio as shownabove is the spectral radius of an n × n matrix. While F ∗ has a larger dimension than the case shown in [1], it isstill a considerable improvement over the n × n matrixthat characterizes the exact decay ratio. From the Perron-Frobenius theorem, the spectral radius also coincides withthe largest eigenvalue of F . We further note that the results in[1] continues to hold if the acceptance rate is homogeneousacross all the nodes, i.e., π i = π, ∀ v i ∈ V . (cid:3) IV. S
IMULATION R ESULTS
In this section, we illustrate the evolution of the epidemicstates in both A-SIYS and A-SIS epidemic models.
A. Impact of transition rate ν in A-SIYS Epidemic We first show the impact of asymptomatic carriers on theepidemic prevalence in the A-SIYS epidemic.
Example 1:
We consider a set of n = 50 nodes and foreach node set the rate of infection β x = β y = 0 . , rateof recovery δ x = δ y = 0 . , activity rate = . , adaptationfactor = . , acceptance rate = . and degree m = 8 . Weinitialize with nodes being susceptible, nodes beingasymptomatic and nodes being symptomatic and simulatethe A-SIYS epidemic till time steps and independentruns. We show the evolution of the fraction of susceptible,asymptomatic and symptomatic nodes averaged over runsin Figure 2 for three different values of the transition rate ν = 0 . , . , . . The upper bound on the decay ratio forthese settings are . , . , . , respectively.Recall that ν captures the rate at which asymptomaticnodes become symptomatic. Given the above parameters,the adaptation and acceptance rates are negligible for symp-tomatic nodes. Therefore, as ν increases, we anticipate thatnodes remain asymptomatic for a much shorter period of timeand consequently the decay ratio will be small. For smallvalues of ν , nodes tend to remain asymptomatic for a longerperiod of time during which they continue to activate andconnect at the same rate as a susceptible node; consequently,the epidemic sustains in the population. Figure 2 shows thatour model captures the above phenomenon. (cid:3)
200 4000 . . . P r o p o r t i o n ν = 0 . , ρ ( A ) = 1 . SXY . . . . P r o p o r t i o n ν = 0 . , ρ ( A ) = 1 . SXY .
51 Time P r o p o r t i o n ν = 0 . , ρ ( A ) = 1 . SXY
Fig. 2: Evolution of proportion of susceptible (S), asymptomatic (X) and symptomatic (Y) nodes in the A-SIYS epidemicaveraged across runs with parameters described in Example 1; ν denotes the rate at which an asymptomatic node becomessymptomatic and ρ ( A ) is the bound on the decay ratio. ,
000 1 , . . . F r a c t i o n I n f ec t e d ρ ∗ = 1 . , ρ p = 1 . ,
000 1 , . . . . F r a c t i o n I n f ec t e d ρ ∗ = 1 . , ρ p = 0 . ,
000 1 , F r e q u e n c y ρ ∗ = 1 . , ρ p = 0 . ,
000 1 , . . . . F r a c t i o n I n f ec t e d ρ ∗ = 0 . , ρ p = 0 . ,
000 1 , . . . . F r a c t i o n I n f ec t e d ρ ∗ = 1 . , ρ p = 0 . , F r e q u e n c y ρ ∗ = 1 . , ρ p = 0 . Fig. 3: Evolution of fraction of infected nodes averaged across runs and histogram of time till the epidemic persists withparameters described in Example 2. B. Evolution of the A-SIS epidemic and comparison with [1]
We now numerically illustrate that the bound obtainedin Theorem 3 above better captures the evolution of theepidemic (fraction of infected population) compared to thebound on the decay ratio obtained in [1] which we denoteby ρ p . We consider two settings; one where ρ p is smallerthan ρ ∗ and second where ρ p is larger than ρ ∗ as describedin the following two examples, respectively. Example 2:
We consider a set of n = 50 nodes and set theinfection rate β = 0 . , recovery rate δ = 0 . and m = 5 .We assume that out of nodes have activity rate a i =0 . , adaptation factor χ i = 0 . and acceptance rate π i =0 . . For the remaining nodes, we choose a i = 0 . and π i = 0 . and vary the adaptation parameter which resultsin varying values of ρ ∗ and ρ p . We initialize with nodesbeing infected and nodes being susceptible and simulatethe A-SIS epidemic. The average fraction of infected nodes across the independent runs for a duration of timesteps is reported in Figure 3. The bounds ρ ∗ and ρ p are shownin the titles of the plots. If at a given point of time (beforethe maximum time-step 1500), all nodes are susceptible (i.e.,the underlying Markov chain has reached the disease-freeabsorbing state) then the simulation ends.The plot on the top left panel of Figure 3 corresponds tothe case with χ i = 0 . for the second group of nodesand shows that the epidemic sustains in the population. Forother cases, the bounds are smaller and it results in the statesreaching the disease-free absorbing state of the dynamics. Wealso plot the histogram of the time the simulation ends for χ i = 0 . and χ i = 0 . (for the second group of nodes)over the runs in the right panel of Figure 3. We note thatfor these two cases, the epidemic sustains in the populationdespite the upper bound on the decay ratio obtained in [1]being smaller than . In contrast, when ρ ∗ < (bottom left
500 1 ,
000 1 , .
51 Time F r a c t i o n I n f ec t e d ρ ∗ = 1 . , ρ p = 1 . ,
000 1 , .
51 Time F r a c t i o n I n f ec t e d ρ ∗ = 1 . , ρ p = 1 . ,
000 1 , .
51 Time F r a c t i o n I n f ec t e d ρ ∗ = 1 . , ρ p = 1 .
50 100 1500510 Time till epidemic sustains F r e q u e n c y ρ ∗ = 1 . , ρ p = 1 . , F r e q u e n c y ρ ∗ = 1 . , ρ p = 1 . ,
000 1 , F r e q u e n c y ρ ∗ = 1 . , ρ p = 1 . Fig. 4: Evolution of fraction of infected nodes averaged across runs and histogram of time till the epidemic persists withparameters described in Example 3.panel), the epidemic reaches the absorbing state in less than iterations in all the independent runs. (cid:3) The above example shows that the epidemic sustains in thepopulation even when ρ p < (but ρ ∗ > ). In the followingexample, we consider parameters with ρ ∗ < ρ p and showthat the epidemic reaches the disease-free state much fasterwhen ρ ∗ is close to even when ρ p is relatively large. Example 3:
We consider a similar setting as above witha set of n = 50 nodes and set the rate of infection β = 0 . ,the rate of recovery δ = 0 . and m = 5 . We assume that out of nodes have activity rate a i = 0 . , adaptation factor χ i = 0 . and acceptance rate π i = 0 . . For the remaining nodes, we choose a i = 0 . and π i = 0 . and vary theadaptation parameter (three values with χ i = 0 . , . and . ) which results in varying values of ρ ∗ and ρ p .We initialize with all nodes being infected and simulate theA-SIS epidemic times. The average fraction of infectednodes across the independent runs for a duration of time steps and the histogram of the time the simulationends are shown in the top and bottom panel of Figure 4,respectively. As the bounds increase, the epidemic sustainsfor a longer time period. However, despite a relatively largevalue of ρ p (but with ρ ∗ closer to ), the epidemic does notsustain for the entire duration in all the simulations. (cid:3) A stark contrast in results can be observed in the top leftpanel of Figure 3 and the top right panel of Figure 4; in theformer, the epidemic sustains for ρ p = 1 . while in thelatter it reaches the disease-free state in most runs even when ρ p = 1 . . To summarize, in both the examples consideredabove, the bound ρ ∗ derived in our work better captures theevolution of the A-SIS epidemic. V. D ISCUSSION AND C ONCLUSION
In this paper, we propose a new activity-driven adaptiveepidemic model that includes asymptomatic carriers presentin several infectious diseases. In the proposed model, symp-tomatic individuals reduce their activation and acceptanceprobabilities while asymptomatic individuals do not, poten-tially because they are not aware of being infected. We showthat the proposed model captures several existing epidemicmodels as special cases. We derive a linearized dynamics thatupper bounds the exact Markovian evolution by resorting toa mean-field approximation. We also highlight a potentialinaccuracy in the upper bound on the decay ratio derived in[1] for the A-SIS epidemic model and generalize their results.The simulation results illustrate that the bound derived in ourwork better captures the evolution of the A-SIS epidemiccompared to the bound obtained in [1].Our work is an early attempt to develop an epidemic modelwith asymptomatic carriers on temporal networks. There areseveral promising avenues for future research in this context.The condition based on the decay ratio is only sufficientto guarantee that the disease is quickly eradicated from thepopulation. In contrast, in the classical SIS epidemic model,when the decay ratio is larger than , there exists a uniqueendemic state that serves as an equilibrium of the mean-field dynamics. While we conjecture that the A-SIS andA-SIYS epidemic models would have a similar behavior,an analogous result has not yet been formally established.Similarly, developing scalable centralized and decentralizedprotection schemes for containing the epidemic in large-scalenetworks in presence of asymptomatic carriers is yet anotherchallenging open problem. EFERENCES[1] M. Ogura, V. M. Preciado, and N. Masuda, “Optimal containment ofepidemics over temporal activity-driven networks,”
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