Prevention of infectious diseases by public vaccination and individual protection

Transcription

1 Noname manuscript No. (will be inserte by the eitor) Prevention of infectious iseases by public vaccination an iniviual protection Xiao-Long Peng Xin-Jian Xu Michael Small Xinchu Fu Zhen Jin Receive: ate / Accepte: ate Abstract In the face of serious infectious iseases, governments eneavour to implement containment measures such as public vaccination at a macroscopic level. Meanwhile, iniviuals ten to protect themselves by avoiing contacts with infections at a microscopic level. However, a comprehensive unerstaning of how such combine strategy influences epiemic ynamics is still lacking. We stuy a susceptibleinfecte-susceptible epiemic moel with imperfect vaccination on ynamic contact networks, where the macroscopic intervention is represente by ranom vaccination of the population an the microscopic protection is characterise by susceptible iniviuals rewiring contacts from infective neighbours. In particular, the moel is formulate both in populations without an then with emographic effects (births, eaths, an migration). Using the pairwise approximation an the probability generating function approach, we investigate both ynamics of the epiemic an the unerlying network. For populations without emography, the emerging egree correlations, bistable states, an oscillations emonstrate the combine effects of the public vaccination program an iniviual protective behavior. Compare to either strategy in Xiao-Long Peng Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 36, China xlpeng@sxu.eu.cn Xin-Jian Xu Department of Mathematics, Shanghai University, Shanghai 2444, China xinjxu@shu.eu.cn Michael Small School of Mathematics an Statistics, The University of Western Australia, Crawley, WA 69, Australia Mineral Resources, CSIRO, Kensington, WA 6151, Australia michael.small@uwa.eu.au Xinchu Fu Department of Mathematics, Shanghai University, Shanghai 2444, China xcfu@shu.eu.cn Zhen Jin Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 36, China jinzhen@sxu.eu.cn

2 2 Xiao-Long Peng et al. isolation, the combination of public vaccination an iniviual protection is more effective in preventing an controlling the sprea of infectious iseases by increasing both the invasion threshol an the persistence threshol. For populations with aitional emographic factors, we investigate temporal evolution of infecte iniviuals an infectious contacts, as well as egree istributions of noes in each class. It is foun that the isease spreas faster but is more restricte in scale-free networks than in the Erös-Rényi ones. The integration between vaccination intervention an iniviual rewiring may promote epiemic spreaing ue to the birth effect. Moreover, the egree istributions of both networks in the steay state is closely relate to the egree istribution of newborns, which leas to uncorrelate connectivity. All the results emonstrate the importance of both local protection an global intervention, as well as the emographic effects. Our work thus offers a more comprehensive escription of isease containment. Keywors Infectious iseases Dynamic networks Ranom vaccination Contact rewiring Mathematics Subject Classification (2) 92D3 (epiemiology) 5C82 (small worl graphs, complex networks) 34C6 (qualitative investigation an simulation of moels) 9B15 (network moels, stochastic) 1 Introuction Since complex network moels (Newman 21) were applie to mathematical epiemiology (Keeling an Rohani 27) in such a way that the host population is moelle as a contact network where noes represent iniviuals an links stan for contacts among them the topological structure of the unerlying network has shown a strong impact on the sprea of infectious iseases (Lloy an May 21; Pastor-Satorras an Vespignani 21a; Meyers et al. 23; Keeling an Eames 25; Barrat et al. 27; Meloni et al. 29; Durrett 21). For instance, base on meanfiel approximations, Pastor-Satorras an Vespignani (21b) suggeste that there is no typical epiemic threshol in a scale-free (SF) network which follows a power-law egree istribution P(k) k γ with the power exponent2 < γ 3 in the thermoynamic limit. Otherwise, there is a positive critical value λ c of the transmission rate λ only for γ > 3. It is worth remarking that in a more mathematically rigorous analysis, Chatterjee an Durrett (29) prove that the critical value λ c is also zero for contact processes on ranom networks with power-law egree istributions for any power exponent γ > 3. In other wors, the infection can pervae the whole SF network provie the transmission rate is nonzero. Over the past ecaes, consierable progress has been mae in the stuy of epiemic moels on static networks, where the network topology remains fixe uring the process of isease transmission. Many topological features of the network, such as the heterogeneity (Eames an Keeling 22; Volz 28), egree correlation (Boguñá et al. 23; Moreno et al. 23) an clustering (Miller 29; Volz et al. 211), were explicitly aresse, yet most stuies ignore the impact of human behavior (in response to infection) on the isease (Ferguson 27; Funk et al. 29;

3 Prevention of infectious iseases by public vaccination an iniviual protection 3 Meloni et al. 211; Poletti et al. 212). Such effect is likely to be especially acute when moifications to human behavior causes changes in the structure of the unerlying contact network (Gross et al. 26; Shaw an Schwartz 28; Zanette an Risau-Gusman 28; Segbroeck et al. 21). Most recently, it has been emonstrate that a better unerstaning of the interplay between human behavior an epiemic ynamics woul help esign effective intervention strategies such as vaccination an social istancing policies, an improve the cost effectiveness of isease control (Funk et al. 21; Fenichel et al. 211). It is therefore of great significance to incorporate human behavior in the stuy of network epiemiology (Perra et al. 211; Durham an Casman 212; Wu et al. 212; Bu et al. 213; Xia an Liu 214; Wang et al. 215). In general, there are two scenarios of human behavior to be consiere in the face of a contagious isease. First, at the macroscopic level authorities enact public containment programs such as quarantine or vaccination; secon, at the microscopic level iniviuals aopt self-protective measures against the isease base on their perceive risk of infection (Funk et al. 21; Durham an Casman 212; Mao 211). Vaccination is one of the most effective public policies for preventing the transmission of infectious iseases (Bauch et al. 23; Althouse et al. 21), an to ate there has been a wealth of stuies on various vaccination strategies for epiemic moels in complex networks (Takeuchi an Yamamoto 26; Shaban et al. 28; Rushmore et al. 214), inicating powerful effects of vaccination on the inhibition of isease transmission, even though some vaccines provie temporary immunity an only reach part of the population. For instance, targete vaccination towars highegree noes is wiely known to have a much higher effectiveness than ranom vaccination either in SF networks (Pastor-Satorras an Vespignani 22) or in small-worl networks (Zanette an Kuperman 22). However, targeting the highestegree noes requires full knowlege of the egree of each noe in the network. To overcome this shortcoming, an alternative vaccination strategy base on local information has been propose (Cohen et al. 23). Uner this strategy vaccination is implemente on a ranom acquaintance of a ranomly chosen noe. Again, this acquaintance vaccination proves to be more effective than ranom vaccination since it prefers to target high-egree noes (this is a consequence of the so-calle frienship paraox on average neighbours of a ranom noe have more neighbours than the noe itself). On the other han, appropriate an timely response of iniviuals to a spreaing contagion may improve the efficiency of vaccination an save meical resources. Several stuies looking at this hypothesis have been frame in network epiemic moels through the aition of a rewiring mechanism (for etails, see Gross an Blasius 28; Schwartz an Shaw 21; Juher et al. 213 an the references therein). A susceptibleinfecte-susceptible (SIS) epiemic moel on aaptive networks was first propose by Gross et al. (26), where at a given rate susceptible iniviuals rewire their connections from infecte neighbours to the susceptibles. The basic iea of this aaptive network epiemiology is that iniviuals ten to change their habits an social contacts to avoi infection in response to a isease threat (Schwartz an Shaw 21). Still further rewiring scenarios have also been consiere reflecting ifferent iniviual reactions to the infectious status of others. For example, Volz an Meyers (27) introuce a neighbour exchange moel to account for the impacts of the stability of

4 4 Xiao-Long Peng et al. social contacts on the sprea of infectious iseases (Rea et al. 28). Alternatively, a ranom swapping of contact partners between susceptible-infecte an infecteinfecte links was stuie in Kamp (21). Most aaptive epiemic moels have largely focuse on the stanar SIS moel (Anerson an May 1992), where the rewiring operation isolates infective iniviuals from the susceptible population in the long run, an therefore influences the epiemic propagation. In turn, the larger the infecte fraction in the population, the more likely susceptible iniviuals will choose to rewire their contacts with infective neighbours. In other wors, the epiemic prevalence can affect iniviual behavior, while avoiing contacts with infection also exerts a strong impact on epiemic ynamics. Typically, in an aaptive network moel both the ynamics of an on the network form a feeback loop (Gross an Blasius 28), which usually involves many interesting ynamical features such as bistability an oscillation (Marceau et al. 21; Gross et al. 26; Zanette an Risau-Gusman 28; Juher et al. 213). So far, little attention has been pai to the combine effects of public vaccination an iniviual protection. Only recently have Shaw an Schwartz aopte a ranom vaccination strategy in an aaptive SIS moel to examine the interplay between iniviual rewiring of social contacts an ranom vaccination of susceptible iniviuals (Shaw an Schwartz 21). In their moel, vaccinate iniviuals secure complete protection from infection unless they return to the susceptible state. That is, the vaccinate can not be infecte irectly from their infective neighbours. However, there is clear evience that some vaccines are not completely efficient, namely, they come with no guarantee of perfect protection from infection (Ganon et al. 21; Smith 22) rather, they confer only partial immunity an a lower transmission rate than the unvaccinate susceptibles. Moreover, in reality, vaccines rarely cover the entire population an only provie finite-time immunity against infection. In light of this, Kribs-Zaleta an Velasco-Hernánez (2) propose a compartmental SIS moel with imperfect vaccination, where vaccinate iniviuals may contract infection irectly from their infectious contacts, although at a vaccine-reuce transmission rate (Brauer 24). Hereafter, we refer to this as the SIV moel. Most recently, this imperfect vaccination moel has been restuie on static SF networks, revealing surprising epiemic behavior (Peng et al. 213). However, the role of iniviual behavior has been neglecte in changing the transmission routes of the isease as well as in reshaping the contact patterns of the unerlying network. This paper aims to provie a systematic framework that couples public vaccination an iniviual protection. For this purpose, we evelop an aaptive SIV moel by extening the compartmental SIV moel (Kribs-Zaleta an Velasco-Hernánez 2) to an aaptive network (Gross et al. 26). The aaptive SIV moel integrates contact rewiring at the microscopic level with vaccination intervention at the macroscopic level in response to isease propagation. In particular, we consier a population without an with emographic effects, respectively, to account for two ifferent cases as follows: case (1) the isease spreas through the population much faster than the births an eaths of iniviuals, an case (2) the isease spreas on a timescale comparable to the spee of iniviuals births an eaths. This observation mainly epens on what type of the isease is being moelle. For instance, the seasonal influenza can be moelle well in a population without emography. Converse-

5 Prevention of infectious iseases by public vaccination an iniviual protection 5 (a) Epiemic spreaing (noe ynamics rule): V S I (b) Aaptive rewiring ofsi-links (link ynamics rule): V S S I V S S I or V S S I Fig. 1 (Colour online) Schematic illustration of the aaptive SIV moel. The moel evolves concurrently accoring to the noe ynamics rule (a) an the link ynamics rule (b) with aaptive rewiring ofsi-links. At each time step, susceptible noes are infecte by their infecte neighbours at a transmission rate α per SI-link an are vaccinate at a per capita rate ϕ. Infecte noes recover an return to being susceptible at a per capita rate β. Vaccinate noes become susceptible at a per capita rate ψ an are infecte at a vaccinereuce transmission rate δα per IV -link ue to imperfect immunity. With the backgroun emographic changes taken into consieration, we assume that each noe can give birth to a susceptible noe with a birth rate η 1, an each noe ies at a natural eath rate η 2. In aition, the infecte noes ie with an extra rate µ because of infection, namely, each infecte noe will ie at a rate η 2 + µ. In the aaptive SIV moel, we assume that only susceptible noes are permitte to rewire connections away from their infecte neighbours at a rate ω per SI-link. In other wors, with rate ω, for each SI-link, the susceptible noe breaks the link to the infecte noe an forms a new link to a ranomly selecte susceptible or vaccinate noe. ly, for the human immunoeficiency virus (HIV) where infection span ecaes, then the emographic impact shoul be taken into account (Keeling an Rohani 27). We establish a mathematical configuration for both cases, which allows for an inepth analysis of the mutual interaction between isease propagation an network evolution. In Sect. 2 we introuce the aaptive SIV moel, an construct a set of approximating eterministic equations for both a close system an one with emography. Section 3 contains main results an Sect. 4 conclues. 2 Aaptive SIV Moel We employ iniviual rewiring of contacts with infective neighbours as an iniviualise self-protective behavior an the vaccination scheme (reucing the rate of infection) as a public intervention in reaction to an infectious isease. The noes of the network represent iniviuals in the population an links are potentially infectious contacts among them. In the aaptive SIV moel, each noe may have only one of the

6 6 Xiao-Long Peng et al. three possible states: susceptible (S), infecte (I) an vaccinate (V ). As emonstrate in Fig. 1(a), the transition probabilities for noe states are as follows: a susceptible noe becomes infecte with the transmission rate α per SI-link (where the parameters in the moel efinition are liste in Table 1); an infective noe recovers back Table 1 Notation use in the moel efinition. Term Meaning α transmission rate per SI-link β recovery rate from infection ϕ vaccination rate of susceptibles ψ relapse rate of vaccinate iniviuals (i.e., each vaccinate returns to being susceptible after an average time perio of τ = 1/ψ ue to temporary immunity) 1 δ effectivity of the vaccine-inuce protection against infection (i.e., vaccinate iniviuals can be infecte at a vaccine-reuce transmission rate δα) η 1 birth rate η 2 natural eath rate µ eath rate from infection ω rewiring rate persi-link where the susceptible noe breaks away from the infecte an then establishes a connection with a ranomly chosen noninfecte noe to the susceptible state with the recovery rate β; accoring to ranom vaccination strategy, a susceptible noe is vaccinate at the vaccination rate ϕ; a vaccinate noe returns to the susceptible state with the resusceptibility rate ψ as the vaccine wears off; ue to imperfect immunity, the vaccinate noe will be infecte irectly from one of its infective neighbours with a reuce transmission rate δα, where the parameter δ measures the inefficacy of the vaccine-inuce protection against infection (δ = means the vaccine is completely effective, while δ = 1 means the vaccine is totally ineffective). With the backgroun emography taken into consieration, we assume that each noe can give birth to a susceptible noe with a birth rateη 1, an each noe (irrespective of its state) ies at a natural eath rateη 2. In aition, the infecte noes ie with an extra rate µ because of infection. That is to say, each infecte noe will ie at a rateη 2 +µ. Meanwhile, the network is rewire aaptively uring the isease propagation [see Fig. 1(b) as an illustration]. For simplicity, only susceptible noes are allowe to rewire their connections from infecte neighbours to ranomly selecte noninfecte (either susceptible or vaccinate) noes with the rewiring rate ω. Self links an multiple links are prohibite. The aaptive SIV moel therefore consists of two processes (Fig. 1): one is the noe ynamics ue to the epiemic spreaing an the birth an eath process, the other is the link ynamics ue to the rewiring of SI-links of the unerlying network. In what follows, we consier two types of population system a close system without emography an an open system with emography, respectively, to account for the following two ifferent cases: case (1) the isease spreas sufficiently fast that the backgroun emographic changes are not significant, an case (2) the epiemic occurs on a timescale comparable to the spee of the birth an eath process. Therefore, some fast iseases like the seasonal influenza can be well escribe with the case (1) moel, whereas for some slow iseases such as HIV we nee to consier the case (2) moel.

7 Prevention of infectious iseases by public vaccination an iniviual protection 7 Table 2 Notation use in the moel formulation an analytical approximation. The subscripts A, B an C in this table inicate the noe state, i.e., A,B,C {S,I,V}, an the lower-case character a (a {s, i, v}) enotes the fraction of noes in state A (A {S, I, V}) corresponingly. Term Meaning A k number of noes in state A with egree k N A = k A k number of noes in state A N k = A A k number of noes with egree k N = k N k = A N A total number of noes in the network a = N A /N fraction of noes in state A, where a = s,i,v corresponing to A = S,I,V, respectively P(k) = N k /N egree istribution of the network (i.e., probability that a ranomly selecte noe has a egree k) k = k kp(k) = 2E/N average egree of the network p Ak (t) = A k /N A probability for a noe in state A to have a egree k g A (x,t) = k p Ak(t)x k probability generating function of p Ak (t) g A (x,t) = x g A(x,t) partial erivative of function g A (x,t) with respect to x g A 2 (x,t) = x 2 g A (x,t) secon-orer partial erivative of function g A (x,t) with respect to x k A = k kp Ak(t) = g A (1,t) average egree of noes in state A p k egree istribution of newborns (i.e., probability for a noe entering the network to have k links) ḡ(x) = k p kx k probability generating function of p k ḡ (x) = xḡ(x) first-orer erivative of function ḡ(x) with respect to x AB-link a link connecting a noe in state A with a noe in state B M AB number of AB-links M A = k ka k = g A (1,t)N A number of links emanating from noes in state A E = ( A M AA + A,B M AB)/2 total number of links in the network P AB = M AB /E fraction of AB-links in the network ABC-type triple a triple in which a noe in state A is connecte to a noe in state B who in turn is connecte to a noe in state C M ABC number of ABC-type triples in the network 2.1 Case (1): Aaptive SIV moel without emography In this case, obviously the emographic turnover can be ignore by setting the emographic parameters to be zero, i.e.,η 1 = η 2 = µ =. There is neither the aition of new noes nor the removal of ol noes, an the rewiring mechanism conserves the total number of links among the entire network. Therefore, the numbers N (of noes) ane (of links) are constant uring the moel evolution. Following Gross et al. (26) an Shaw an Schwartz (21), we also evelop a mean-fiel moel for the system, which tracks the ynamics of both noes an links. The notation summarise in Table 2 allows us to obtain the following set of ifferential equations for the evolution of the fractions of noes as follows: t s = βi+ψv ϕs αe N P SI, (1) t i = αe N P SI +δα E N P IV βi, (2) t v = ϕs ψv δαe N P IV. (3)

8 8 Xiao-Long Peng et al. On the right han sie (rhs) of Eq. (1), the first an secon terms correspon to the conversion of infecte an vaccinate noes into susceptible ones, respectively, while the thir term inicates the vaccination process of susceptible noes, an the fourth term represents the loss of susceptible noes ue to infection, which is proportional to the numberep SI ofsi-links an the transmission rate α per SI-link. On the rhs of Eq. (2), the first an secon terms consier that infection spreas along each SI-link with rate α an along each IV -link with rate δα, respectively, while the thir term escribes recovery of infecte noes at rateβ. Similarly, the first term on the rhs of E- q. (3) escribes the gain of vaccinate noes ue to vaccination of susceptible noes, while the secon an thir terms escribe the loss of vaccinate noes as a result of immunity wearing off an infection along each IV -link, respectively. The noal ynamics interact closely with the ynamics of links through the change of noal states an infection of noninfecte noes along SI- an IV -links. In aition, rewiring of SI-links also implicitly affects the system since it alters the numbers of SI-, SS-, an SV -links. To close the moel system, we use the low-orer moment closure approximation M ABC M AB M BC /N B (Keeling et al. 1997) with the coefficient ( k 1)/ k being omitte, where A,B,C {S,I,V} an M ABC represents the number of ABC-type triples in which a noe in state A is connecte to a noe in state B who in turn is connecte to a noe in state C. This leas to the following equations for the evolution of the fractions of links: t P SS = βp SI +ψp SV +ω s s+v P SI 2ϕP SS 2α E P SI N s P SS, (4) t P SI = 2α E P SS P SI +2βP II +(ψ +δα E P SV N s N v )P IV (α+β +ϕ+ω +α E P SI N s )P SI, (5) t P SV = βp IV +2ψP VV +2ϕP SS +ω v s+v P SI (α E P SI N s +δαe P IV +ϕ+ψ)p SV, N v (6) t P II = α(1+ E P SI N s )P SI +δα(1+ E P IV N v )P IV 2βP II, (7) t P IV = α E N (P SIP SV +2δ P IVP V V )+ϕp SI (δα+δα E P IV +β +ψ)p IV, s v N v (8) t P V V = ϕp SV 2ψP VV 2δα E P IV P VV, N v (9) where the triple approximations in terms of the number of pairs have been absorbe into this formulation. The first three terms on the rhs of Eq. (4) escribe the gain of SS-links as a result of recovery, loss of immunity, an rewiring, respectively, while the last two terms correspon to the loss of SS-links ue to vaccination of susceptible noes an infection through SSI-type triples, respectively. Equations (5) (9) are analogous except that infection of vaccinate noes at a reuce rate δα has been taken into account. Since the numbersn (of noes) ane (of links) are constant in this moel, the whole system is close uner the conservation conitionss+i+v = 1 anp SS +P SI +P SV +P II +P IV +P VV = 1. The steay states of the above orinary ifferential equations (1) (9) can be tracke as a function of parameters via the continuation software AUTO (27), which helps investigate the ynamical effects of rewiring an vaccination through the bifurcation analysis (see the Results section 3).

9 Prevention of infectious iseases by public vaccination an iniviual protection Case (2): Aaptive SIV moel with emography In the previous section, we assume that the time scale of epiemic sprea is much faster than the time scale of population births an eaths. This is not aequate for slow iseases which last for ecaes. To overcome this shortcoming, we take into account the backgroun emographics renewal in the aaptive SIV moel, where (i) each iniviual can give birth to a susceptible at a birth rate η 1 an the newborn ranomly connects to k existing iniviuals accoring to a preassigne birth egree istribution p k, which correspons to the probability generating function ḡ(x), an meanwhile (ii) each iniviual ies from natural causes at a rate η 2 an the infective ies at an extra rate µ (possibly, one expects µ > ue to the isease itself something which is omitte from SIS/SIV moels without emography). We o not consier the possibility of either vaccination or infection being passe irectly to offspring. The low-imensional mean-fiel equations in the case (1) allows us to carry out in-epth bifurcation analysis an preict rich ynamical features for the close system. However, the case (1) moel is base on a homogeneous mixing assumption an low-orer moment-closure approximations, which funamentally neglect the effects of the unerlying network structure, generally resulting in inaccurate preictions about the time evolution of the system (Marceau et al. 21). One important network property to be consiere is the heterogeneity of the egree istribution among noes (Pastor-Satorras an Vespignani 21a,b). To make our moel applicable to heterogeneous networks, the network noes are assemble into subgroupsa k accoring to their egreek, wherea k enotes the number of noes in state A {S,I,V} withk connections. With the notation liste in Table 2, one can write the evolution equations for the numbers of susceptible, infecte an vaccinate noes withk connectionss k, I k, anv k as follows: t S k = η 1 N p k η 2 S k α M SI M S ks k ϕs k +η 1 ḡ (1)(S k 1 S k )+βi k +ψv k +ωm SI S k 1 S k N S +N V η 2 [ks k (k +1)S k+1 ] µ M SI M S [ks k (k +1)S k+1 ], (1) t I k = (η 2 +µ)i k +α M SI M S ks k +δα M IV M V kv k +η 1 ḡ (1)(I k 1 I k ) βi k ω M SI M I [ki k (k +1)I k+1 ] η 2 [ki k (k +1)I k+1 ] µ 2M II M I [ki k (k +1)I k+1 ], (11) t V k = η 2 V k +ϕs k δα M IV M V kv k +η 1 ḡ (1)(V k 1 V k ) ψv k +ωm SI V k 1 V k N S +N V η 2 [kv k (k +1)V k+1 ] µ M IV M V [kv k (k +1)V k+1 ]. (12) On the rhs of Eq. (1), the first (secon) term enotes the creation (loss) ofs k ue to natural births (eaths), whereη 1 N is the increasing rate of newborn susceptibles that are of egreek with probability p k (accoring to the egree istribution of newborns). The thir term correspons to the loss ofs k because of infection that is proportional

10 1 Xiao-Long Peng et al. to the number of connections k, the transmission rate per SI-link α, an the probabilitym SI /M S for a link originating from the susceptible to point to the infective, which is assume to have no egree correlation with the starting noe. The fourth term represents the removal of S k ue to vaccination. The fifth term consiers a growth rate η 1 N of newborns, each of which on average has a number k = ḡ (1) = k k p k of links. The number S k increases (ecreases) when the susceptibles with egree k 1 (egreek) are connecte to by one of the η 1 Nḡ (1) links emanating from the newborn noes. In aition, the probability for the newborns to ranomly link to the susceptibles with k contacts is S k /N. Note that the establishment of connections of the newborns is simply base on the ranom attachment mechanism. The sixth an seventh terms refer to the recovery process from infection an the relapse to susceptibility for vaccinate noes as the vaccine wears off, respectively. The eighth term is attributable to the rewiring process with the rewiring rateω persi-link. The number S k increases (ecreases) when the susceptibles with egree k 1 (egree k) is ranomly rewire towars by the susceptible enpoint on the SI-link after isconnecting from the infecte enpoint. An, the probability for the susceptible enpoint to ranomly rewire to the susceptibles with egree k from the pool of noninfecte noes iss k /(N S +N V ). The ninth term epicts the changes ofs k owing to contacts lost from naturally ying noes. The number S k increases (ecreases) when one of the k + 1 neighbours (k neighbours) of the susceptible with egree k + 1 (egree k) ies naturally at a rateη 2. The tenth term escribes the changes ofs k ue to contacts lost from noes ying from infection. The numbers k increases (ecreases) when one of the infecte neighbours of the susceptibles with egree k + 1 (egree k) ies from infection at a rate µ. There is a fraction M SI /M S of infecte noes among the ks k neighbours of the susceptibles with egree k. Here, we assume that the probability for a link emanating from a susceptible noe to point to an infecte noe is uncorrelate to the egree k of the starting noe. The erivation of Eqs. (11) an (12) can be explaine analogously. The evolution equations for the numbers of susceptible (N S ), infecte (N I ), an vaccinate (N V ) noes, as well as the numbers of links emanating from susceptible (M S ), infecte (M I ), an vaccinate (M V ) noes, can be erive accoring to Eqs. (1) (12), as given in etail in Appenix A. To close the system an to take into account the ynamical correlations in the infection status of connecte iniviuals, aitional equations for the evolution of the number M AB of AB-links are require, where A,B {S,I,V}. In a similar

11 Prevention of infectious iseases by public vaccination an iniviual protection 11 manner, we obtain t M SS = η 1 ḡ (1)N S +βm SI +ψm SV +ωm SI 2η 2 M SS 2ϕM SS N S +N V N S 2M SS g S αm (1,t) SI (13) M S (1,t), g S t M SI = η 1 ḡ 2M SS g S (1)N I +2βM II +ψm VI +αm (1,t) SI M S g S (1,t) +δαm IV M SV g V (1,t) M V g V (1,t) M SI g S (2η 2 +µ+α+β +ϕ+ω)m SI αm (1,t) SI (14) M S (1,t), t M SV = η 1 ḡ N V (1)N V +βm IV +2ϕM SS +2ψM VV +ωm SI (2η 2 +ϕ+ψ)m SV N S +N V M SV g S αm (1,t) SI M S g S (1,t) δαm M SV g V (1,t) IV (15) M V (1,t), t M M SI g S II = αm SI +δαm IV +αm (1,t) SI M S g S (1,t) +δαm IV t M M SV g S IV = ϕm SI +αm (1,t) SI M S g S (1,t) +δαm IV g V 2M VV g S M IV g V (1,t) M V g V (1,t) 2(η 2 +µ+β)m II, (16) g V (1,t) M V g V (1,t) (2η 2 +µ+δα+β +ψ)m IV M IV g V δαm (1,t) IV (17) M V (1,t), g V t M 2M VV g V VV = ϕm SV 2(η 2 +ψ)m VV δαm (1,t) IV (18) M V (1,t). On the rhs of Eq. (13), the first four terms escribe the aition of SS-links as a result of newborn noes (with an average connectivity k = ḡ (1)) connecting to ol susceptible ones with probability N S /N, the recovery of the infecte enpoints on SI-links at rate β, the resusceptibility of vaccinate enpoints of SV -link ue to temporary immunity, an, the rewiring of SI-links away from the infecte noes an towars other susceptible ones with probabilityωn S /(N S +N V ), respectively; while the last three terms correspon to the loss of SS-links ue to the natural eath (at a rate 2η 2 ) an the vaccination (at a rate 2ϕ) of one of the two enpoints of SSlinks, an the infection (at a rate α) of the intermeiate susceptible noe along the SSI-type triples, respectively. The appearance of the probability generating functions in Eq. (13) represents an improvement in the approximations of the number of triples in terms of the number of pairs by taking into account the heterogeneity of the central noe s egree k, rather than simply assuming a homogeneous mixing. A complete escription of this type of formulation is given in Appenix B. Explanations for the erivation of Eqs. (14) (18) are similar except that the infection of vaccinate noes at a vaccine-reuce transmission rate δα has also been consiere. Equations (13) (18) characterise the contact behavior among iniviuals of ifferent states an how they change in response to epiemic transmission an aaptive rewiring of SI-links. In conjunction with the set of Eqs. (1) (12), they govern the g V

12 12 Xiao-Long Peng et al. temporal evolution of both the epiemic an the unerlying contact network. Our analytic approach in Eqs. (1) (18) improves on the usual heterogeneous mean-fiel approximation by Pastor-Satorras an Vespignani (21a,b) as we take into account both the connectivity heterogeneity but also the ynamical correlation among the infection states of ajacent noes, which is generally ignore in the heterogeneous mean-fiel approximation (Gleeson et al. 212). It is worth noting that Marceau et al. (21) has recently propose an improve heterogeneous moment-closure approximation, referre to as the active neighbourhoo approach, to stuy the SIS epiemic ynamics on aaptive networks. This active neighbourhoo approach categorises all the noes base on the number of their susceptible an infecte neighbours. In the approximation for the number of triples, our approach only takes the central noe s egree into consieration, but the active neighbourhoo approach by Marceau et al. (21), by contrast, takes the starting noe s an the central noe s egrees (both the susceptible egree an infectious egree) into account. In this regar, our approach may lose some accuracy for the approximation of number of triples. On the basis of the active neighbourhoo approach, Silk et al. (214) use generating functions with two spatial variables to investigate an aaptive voter moel, proviing new insights into the connection between network science an the theory of partial ifferential equations. Moreover, uner the formalism framework of the active neighbourhoo approximation approach, some improve moment-closure approximations have also been iscusse for aaptive networks(shkarayev an Shaw 213; Demirel et al. 214). Due to the etaile information about the total egree an the infectious egree inclue in the active neighbourhoo approach, the above moment-closure approximations are more accurate but is of much greater computation complexity in comparison with the moment-closure approach as in Appenix B, where the approximation of the number of triples only consiers the central noe s egree. Moreover, all these approximation approaches have assume zero egree correlation among n- oes in the network. 3 Results 3.1 The moel without emography For the aaptive SIV moel without population births an eaths, we focus on the effects of rewiring an vaccination on both ynamics of the epiemic an the network. The steay state of the system of Eqs. (1) (9) can be tracke as a function of parameters with the bifurcation software AUTO (27) Bifurcation structure We first consier the bifurcation structure of the moel. Figure 2 shows the bifurcation iagram of the infective fractioni as a function of the transmission rateα. The analytic results from the mean-fiel Eqs. (1) (9) are compare with computational simulations of the full moel. In the case of no rewiring (ω = ) [Fig. 2(a)], a single transcritical bifurcation occurs at the well-known epiemic threshol (Anerson an May

13 Prevention of infectious iseases by public vaccination an iniviual protection 13 1 i i i (a) (b) (c) III I II II I Fig. 2 (Colour online) Densities of infecte noes i as a function of the transmission rate α for ifferent rewiring rates: ω = (a),.4 (b), an.2 (c), respectively. Black soli (ashe) lines: stable (unstable) branches; blue (green) soli lines: lower (upper) turning point of the stable epiemic cycles; blue (green) ashe lines: lower (upper) turning point of the unstable epiemic cycles. Dots an error bars (if larger than the ots): simulation results. All the simulation points are obtaine by averaging over 5 runs, each of which goes through time steps, starting with an initial fraction.1% of infecte sees an all the others susceptible. Parameters aren = 1 4,E = 1 5, β =.2, ϕ =.8, ψ =.2, an δ =.2.

14 14 Xiao-Long Peng et al. 1992), α c, which inicates the critical value of transmission rate for invasion of new infections. Hereafter we refer to this threshol as the invasion threshol. In this setting,α c =.14, coincies with the critical behaviorα c = β(ϕ+ψ)/( k (δϕ+ψ)) obtaine in Peng et al. (213). If α > α c, the isease-free (healthy) state loses stability while the enemic state becomes stable. As a low rewiring rate ω =.4 is introuce [Fig. 2(b)], the epiemic threshol grows rapily to.284. With an increment in α, a Hopf bifurcation takes place at α =.325 with the appearance of a stable epiemic cycle (Gross et al. 26) (i.e., the limit cycle, which is a perioic solution), an then a region of perioic oscillations is observe in the range of.325 < α <.523. In aition, two sale-noe bifurcations occur at α =.374 an α =.596, respectively, between which a clear region of bistability is establishe, as shown in the ark gray region I (where a stable enemic state an a stable epiemic cycle coexist) an the meium gray region II (with two coexisting stable enemic states). As the rewiring rate ω rises to.2 [Fig. 2(c)], the sale-noe bifurcation on the left inuces an enemic threshol, above which an alreay establishe enemic persists. For convenience, we refer to this threshol as the persistence threshol. Across the turning point of the right sale-noe bifurcation (a little to the right), there is a cycle fol bifurcation (i.e., the fol bifurcation of limit cycles) at α =.212, where stable an unstable epiemic cycles converge, an then isappear. Between the left sale-noe bifurcation an the cycle fol bifurcation, a wier bistable region is locate: in the ark gray region I, a stable enemic state an a stable epiemic cycle coexist; in the meium gray region II, there are two stable enemic states; an in the light gray region III, a stable isease-free state coexists with a stable enemic state. Again, an oscillation phenomenon is observe in the ark gray region I. In our simulations searching for the branch points in the bistable regions in Fig. 2, we have first examine which attractor the system goes to by the continuation software AUTO (27), thereby we obtaine istinctive ranges of ifferent stable branches, an then average over consierable runs that appeare to be in the same basin of attraction. In a similar manner to Shaw an Schwartz (28), we locate the stable enemic state branch by sweeping the transmission rate α from large to small values, an the steay state was average over 5 runs, each of which goes through5 1 4 time steps, starting with an initial fraction.1% of infecte sees. The simulation results that roppe into the range of high stable enemic branch were regare as the high enemic branch point, while the simulation results that roppe into the range of low stable enemic branch were regare as the low enemic branch point. To locate the stable isease-free state branch, we swept α from zero to large. We consiere the isease-free state to be stable if the isease ie out in any run of simulation. However, our metho might fail to fin stable states with small basins of attraction ue to the stochastic nature of computational simulations an because the isease-free s- tate is absorbing. Generally speaking, the simulation an the analytic results in Fig. 2 are in goo agreement, especially in the upper branches (enemic state) an the lower branches (isease-free state). However, in the oscillatory regions where the limit cycles coexist with a stable enemic state, there are a little iscrepancies between theory an simulations. Elsewhere it has been foun that there is often isagreement between mean-fiel (or pairwise) approximation approaches an network simulation-

15 Prevention of infectious iseases by public vaccination an iniviual protection 15 s about the parameter ranges for oscillations. In Fig. 2(b), while we have observe perioic oscillations (limit cycles) in the region I, the error bars suggest there is potential for oscillations to appear also in the region II. On the other han, the error bars in Fig. 2(c) are so small that it is unclear whether oscillations appear in region I as preicte by the mean-fiel approximation. By varying the vaccination rate ϕ an tracking each bifurcation curve for the rewiring rate ω =.4, we obtain the two-parameter bifurcation iagram on the (α,ϕ) plane in Fig. 3(a). In the light green, white, an yellow regions, there is only a single attractor, which is a isease-free state in the light green (Healthy) region, an enemic state in the white (Enemic) region, an an epiemic cycle in the yellow region. The rest of the plane in Fig. 3(a) represents a bistable region, which is ivie into three colours in ifferent gray levels: in the light gray region I both the iseasefree equilibrium an an enemic equilibrium are stable; in the meium gray region II there are two stable enemic equilibria; an in the ark gray region III an enemic equilibrium an an epiemic cycle are both stable. Oscillations can be observe both in the yellow region an the ark gray region III since there exists a stable epiemic cycle. Analogously, we get in Fig. 3(b) the two-parametric bifurcation iagram for a larger rewiring rateω =.2. In the light green (Healthy) region, there is only a stable isease-free equilibrium. Furthermore, there are three ifferent regions of bistability: in the light gray region I, both the isease-free an enemic states are stable; in the meium gray region II, there are two stable enemic states; an in the ark gray region III, a stable epiemic cycle coexists with a stable enemic state. Also, oscillatory ynamics can be observe in the ark gray region III. On the contrary, in the white (Enemic) region, there is only one stable enemic state. From Fig. 3, it can be foun that for ϕ = (i.e., without vaccination), there is a backwar bifurcation an a sale-noe bifurcation. As the vaccination rate ϕ graually increases, the moel exhibits a more complicate bifurcation structure with the occurrence of two sale-noe bifurcations, Hopf bifurcations, oscillations, an the cycle fol bifurcation. Finally, for ϕ sufficiently large, the epiemic cycles isappear an the ynamical system is characterise by a transcritical bifurcation an two sale-noe bifurcations. Moreover, the bifurcation points of both the transcritical an sale-noe bifurcations shift rightwars with the increase of ϕ. In other wors, the public vaccination intervention plays a crucial part in preventing an controlling the sprea of infectious iseases by increasing the invasion threshol an also the persistence threshol for epiemics. Furthermore, in comparison with Fig. 3(a), the isease-free region is larger in Fig. 3(b), which suggests that the larger the aaptive rewiring rateω is, the safer the population is from infection. To unerstan the effect of iniviual rewiring on the epiemic ynamics in more etail, a two-parameter bifurcation iagram on the plane of(α,ω) is plotte in Fig. 4. In the case of no rewiring (i.e.,ω = ), there is only the transcritical bifurcation (enote by a soli line), which separates the healthy an enemic regions. As ω increases, the system first enters a bistability region (at ω =.124), i.e., the meium gray region III (where two stable enemic states coexist), an then unergoes oscillatory ynamics (at ω =.186) with the presence of a stable epiemic cycle associate with a supercritical Hopf bifurcation (see the yellow region). As ω grows further, the

16 16 Xiao-Long Peng et al. vaccination rate (x1-4 ) 4 Healthy 2 I Oscillatory Bistable III Enemic Healthy 2 Bistable I Enemic II (a) II (b) Oscillatory III Enemic Fig. 3 (Colour online) The two-parameter bifurcation iagram on the (α, ϕ) plane for ifferent rewiring rates: ω =.4 (a) an.2 (b). The yellow (in panel (a)), the white an light green regions, respectively, represent a single attractor region, where it is an epiemic cycle in the yellow region (in panel (a)), an enemic state in the white (Enemic) region, an a isease-free state in the light green (Healthy) region. The remaining area represents a bistable region, which is ivie into three colours in ifferent gray levels: in the light gray region I both the isease-free state an an enemic state are stable; in the meium gray region II there are two stable enemic states; an in the ark gray region III an enemic state an an epiemic cycle are both stable. Oscillations can be observe both in the yellow region (in panel (a)) an in the ark gray region III ue to the presence of a stable epiemic cycle. Purple (soli) lines: the transcritical bifurcation points; re (ash-otte) lines: the sale-noe bifurcation points; black (ashe) lines: the Hopf bifurcation points; blue (otte) lines: the cycle fol bifurcation points. Parameters are N = 1 4, E = 1 5,β =.2, ψ =.2, an δ =.2.

17 Prevention of infectious iseases by public vaccination an iniviual protection 17 rewiring rate Healthy Oscillatory III I II IV Bistable Oscillatory Enemic Fig. 4 (Colour online) Two-parametric bifurcation iagram on the (α, ω) plane. In the white, yellow, an light green regions, there is only a single attractor, which is an enemic state in the white (Enemic) region, an epiemic cycle in the yellow region, an a isease-free state in the light green (Healthy) region. The remaining part represents a bistable region, which is ivie into three colors in ifferent gray levels: In the light gray region I both of the isease-free an enemic states are stable; in the meium gray regions II an III there are two stable enemic states; an in the ark gray region IV a stable enemic equilibrium an a stable epiemic cycle coexist. The oscillatory ynamics can be observe in both the yellow an the ark gray regions since there exists a stable epiemic cycle. The purple soli, re ash-otte, black ashe, an blue otte lines, respectively, correspon to the transcritical, sale-noe, Hopf, an cycle fol bifurcation points. Parameters are as in Fig. 2. system enters another bistable region with one stable enemic state an one stable epiemic cycle (see the ark gray region IV), where the oscillatory phenomena can still be observe. For large enough ω, the whole bistable region is ivie by transcritical an Hopf bifurcations into three parts: in the light gray region I, one stable healthy state coexists with one stable enemic state; in the meium gray region II, two stable enemic states coexist; an in the ark gray region IV, one stable enemic s- tate coexists with one stable epiemic cycle. While the increase of ω slightly enlarges the healthy region, it greatly narrows the enemic region an wiens the bistable region. Moreover, the significant role of rewiring in hinering epiemics is clear in the enlarge invasion threshol as well as the emergence of a persistence threshol. Accoring to Figs. 3 an 4, we conclue that the combination of public vaccination an local rewiring is more effective in preventing an controlling the sprea of infectious iseases, although it oes cause complicate (an interesting) ynamical behavior to occur.

18 18 Xiao-Long Peng et al. P(k) (a) (b) P(k).6.3 susceptible infecte vaccinate Fig. 5 (Colour online) Degree istribution in each class of noes for ifferent epiemic prevalences in the steay enemic state: a high epiemic prevalence i =.95 with average egrees of susceptible, infecte, an vaccinate iniviuals k S = 21, k I = 19, an k V = 486 (a); an a low epiemic prevalence i =.53 with k S = 15, k I = 12, an k V = 7 (b). The black (soli), re (ashe), an blue (ash-otte) lines correspon to the susceptible, infecte, an vaccinate iniviuals, respectively. The vaccinate iniviuals have a very broa egree istribution, which extens beyon the omain of the figure. Parameters are N = 1 4, E = 1 5, α =.6, β =.2, ϕ =.8, ψ =.2, δ =.2, an ω =.4. k

19 Prevention of infectious iseases by public vaccination an iniviual protection Network topology We first examine the egree istribution in the enemic steay state. We choose parameters in the bistable region to compare the egree istributions of noes in each class at ifferent enemic states. In particular, the egree istributions at a high epiemic prevalencei =.95 an a low prevalencei =.53 are shown in Figs. 5(a) an 5(b), respectively. The results are obtaine by averaging over time steps at the steay state with 5 inepenent simulations for the moel. As shown in Fig. 5, the average egree of infecte noes is smaller than that of susceptible noes an far smaller than that of vaccinate ones because of rewiring. In the close population without emographic turnover, the breaking of SI-links causes the infective noes to lose susceptible neighbours, while the susceptible an vaccinate noes corresponingly gain more connections. Therefore, the average egree of susceptible noes is larger than the average egree of infecte noes in the steay state. In this regar, the ranom vaccination of the susceptibles favors higher-egree noes in the network ue to rewiring (Shaw an Schwartz 21), an thus is expecte to be effective, as in previous stuies on targete vaccination (Pastor-Satorras an Vespignani 22; Zanette an Kuperman 22). Since the vaccine-reuce transmission rate is relatively low, vaccinate noes accumulate connections from rewiring, therefore they have a broa egree istribution with a very large average egree [see k V = 486 in Fig. 5(a) an k V = 7 in Fig. 5(b)]. There are more SI-links (at a lower level of epiemic prevalence but with a higher ensity of susceptible noes) in Fig. 5(b) than in Fig. 5(a), thus aaptive rewiring has stronger effect on the average egree of infecte noes, which in turn reuces the average egree of susceptible noes ue to the recovery process. It is worth remarking that the presence of the low infection an high infection states is interesting, because the original aaptive network moel by Gross et al. (26) just foun bistability between enemic an isease-free states. In essence, the ifferences between the low infection an high infection states are roote in the ifferent selections of target noes towars which the original susceptible noes will be rewire. Accoring to the rewiring mechanism in our moel, after susceptible noes break away their connections from infecte neighbours, the susceptibles establish new connections towars ranomly selecte noninfecte noes, namely, either susceptible or vaccinate. Note that susceptible noes can be infecte at a much higher rate than vaccinate noes. If vaccinate noes are selecte as the rewiring target, then more vaccinate noes are able to accumulate connection from rewiring. Thus the mean egree of vaccinate noes are much higher an they become hub noes in the network. Among the neighbours of vaccinate noes, susceptible noes are in the majority since they are rewire towars the vaccinate noes. On the other han, vaccinate noes are not permitte to rewire away from infecte neighbours. Once the vaccinate noes are infecte by one of their neighbours, these hub noes will inuce more infections, leaing to a high prevalence in the steay state. On the contrary, if more susceptible noes are selecte as the rewiring target, then the susceptible noes ten to form a cluster an the vaccinate noes to some extent lose the ability of accumulating new connections from rewiring. However, susceptible noes are less