Stochastic epidemic models revisited: Analysis of some continuous performance measures

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1 Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, Madrd, Span A. Economou Department of Mathematcs, Unversty of Athens, Panepstemopols, 5784 Athens, Greece M.J. Lopez-Herrero School of Statstcs, Complutense Unversty of Madrd, Madrd, Span emal: Lherreroestad.ucm.es Abstract We deal wth stochastc epdemc models havng a set of absorbng states. The am of the paper s to study some contnuous characterstcs of the epdemc. In ths sense, we rst extend the classcal study of the length of an outbreak by nvestgatng the whole probablty dstrbuton of the extncton tme va Laplace transforms. Moreover, we also study two almost new epdemc descrptors; namely, the tme untl a non-nfected ndvdual becomes nfected and the tme untl the ndvdual s removed from the nfectve group. The obtaned results are llustrated by numercal examples ncludng an applcaton to a stochastc SIS model for head lce nfectons. Keywords: Extncton tme; Head lce; Recovery tme; Stochastc SIS model; Stochastc SIR model; Tme to nfecton AMS Subject Class caton: 92D25, 92D30 Introducton Understandng the mechansm that underles the spread of an nfectous dsease can gve mportant nsghts to help n the ght aganst the dsease tself. Epdemc models are wdely used for ncreasng the understandng of nfectous dsease dynamcs and for determnng preventve measures to control nfecton spread.

2 In the present work, we descrbe the dynamcs of the epdemc n terms of a general brth-death model (ncludng epdemc SIS models) and the stochastc SIR model. We suppose that a closed populaton s dvded nto susceptble, nfectve and, for SIR models, also removed ndvduals. The assocated process descrbes the composton of the populaton and termnates when the number of nfectves becomes zero, whch almost surely happens wthn nte tme. As a result, the statonary dstrbuton s degenerate on the set of absorbng states. We refer the reader to the textbooks by Allen (2003), Andersson and Brtton (2000) and Daley and Gan (999) n order to nd the mathematcal background of these models as well as the man results and applcatons wthn a bologcal framework. In ths paper, we concentrate manly on three contnuous characterstcs of ths spread. The rst characterstc, the extncton tme, quant es the spread of the epdemc on the whole populaton and descrbes the tme tll the end of the epdemc process. The other two characterstcs concern the ndvduals behavor. More concretely, we deal wth the tme tll the nfecton and the recovery/removal tme of a selected nfected ndvdual. The extncton tme has been the subject matter of many papers. Frst, we focus on the determnaton of the moments n nte brth-death processes. Norden (982) rst obtans an explct expresson for the mean tme to extncton, gven an ntal state. Then, he uses the backward Kolmogorov equatons to get an expresson for hgher moments nvolvng the moments of one order less. In addton, the densty functon of the extncton tme s approxmated n terms of a gamma functon. For an alternatve proof of the moment formulae based on the use of the Laplace transform method, we refer to Goel and Rchter-Dyn (974). For the stochastc logstc epdemc, Krysco and Lefèvre (989) obtan an approxmaton for the mean tme to extncton based on the combnaton of several prevous results. More recently, Newman et al. (2004) derve explct expressons for the mean and the varance of the extncton tme for the specal case of a sngle ntal nfectve. Stone et al. (2008), for an SIS model wth external source of nfecton, deal wth the tme to reach 0 nfectves startng from a certan number of nfected ndvduals and determne expressons for hgher order moments usng the moment generatng functon. The study of the moments can be extended to the n nte case; that s, the case where brth-death process takes values on N (see, for example, Allen (2003) and Renshaw (993)). The problem of determnng the whole dstrbuton of the extncton tme (.e., dstrbuton functon or densty functon) can be nvestgated usng varous methodologes ncludng, for nstance, spectral decompostons (see Kelson (964)) and generatng functon methods (see Norden (982)). At ths pont, we remark that ths problem s much more nvolved than the calculaton of the moments. There s no doubt about the theoretcal value of the exstng results, but ther sutable computaton s an ntrcate matter. Snce the extncton tme can be reduced to the transent analyss of the brth-death process, we next gve a bref overvew of some exstng methods. In the more general framework of a contnuous tme Markov chan on a nte state space wth rate matrx Q, the uncondtonal verson of the absorpton 2

3 tme, L, sats es the followng results (see Kulkarn (995) and Latouche and Ramaswam (999)): (R) P fl xg = expfmxge; for x 0: (R2) If M s nvertble, L s nte wth probablty. Then, we have that '(s) = E[e sl ] = (si M) Me and E[L k ] = k!( M ) k e; for k ; where M s the submatrx of Q correspondng to the set of the N transent states, I s the dentty matrx of order N, s a row vector of dmenson N contanng the ntal probabltes and e s a column vector of dmenson N wth all entres equal to one. Comng back to the context of the nte brth-death process, t s clear that the matrx M s an rreducble dagonally domnant matrx. Therefore, M s nvertble and t s possble to analyze the behavor of transforms and moments of the extncton tme, gven an ntal state, n terms of the results for the uncondtonal absorpton tme, L: Thus, equaton (R) gves a closedform soluton of the problem under study. However, we readly notce that the computaton of the prevous formulae n (R) and (R2) requres to deal wth powers and nverses of matrces havng postve and negatve entres, whch s numercally unstable. Fndng accurate methods to compute the matrx exponental s a non-trval matter whch stll attracts the nterest of many nvestgators n numercal analyss. Followng Moler and Van Loan (2003), we remark that "the exponental of a matrx can be computed n many ways. In practce, consderaton of computatonal stablty and e cency ndcates that some of the methods are preferable to others, but that none are completely satsfactory". One of the best methods s the scalng and squarng method mplemented n MATLAB, whch computes a Padé approxmaton to the matrx exponental (see Hgham (2005)). Kulkarn (995) presents four methods for dealng wth the transent analyss of a nte Markov chan. In addton to the exponental matrx, those methods also nclude d erental equatons, Laplace transforms and unformzaton technques. Some dscusson s gven, but t does not lead to a de nte evdence n favor of one of the methods. In ths paper, we develop recursve schemes for Laplace transforms and appeal to numercal nverson methods. The numercal nverson of Laplace transforms s based on Fourer seres methods. More concretely, our numercal results are obtaned by employng the Post-Wdder method (see Abate and Whtt (995) and Cohen (2007)). A BASIC code mplementng the algorthm can be found n Abate and Whtt (995). Other methods for numercally nvertng Laplace transforms are avalable n the lterature. In fact, t s recommended to perform two d erent numercal nverson algorthms (e.g. Post-Wdder method and Euler method) so that they can be used n parallel to determne the desred accuracy by agreement of the two methods. The possblty of performng the nverson usng varous alternatve methods, as a checkng mechansm, gves an ntal motvaton to use the Laplace transform method. 3

4 At ths pont, we menton that the Laplace transform provdes a methodologcal approach not only for the computaton of the dstrbuton of the extncton tme, but also for the study of the tme to nfecton and recovery of a selected ndvdual, as we wll show n the sequel. However, the study of these two descrptors could also be reduced to an absorpton tme problem by ntroducng an auxlary absorbng state. Transton nto ths state would mean that the marked ndvdual gets nfected (or he gets recovered). A detaled comparson among d erent methods of numercal computaton s not our objectve here, but a few general consderatons are presented n what follows. The varety of numercal methods for computng the matrx exponental makes t d cult to summarze how they should be compared. Moreover, t s nterestng to notce that computng only expfag requres a d erent approach from computng expfaxg; for several values of x. We also observe that when dealng wth Markov chans, the computaton of the matrx exponental s subject to probablty constrants. As a result, a rst obvous remark s that there s no unque concluson about whch method s the best and t probably depends on each concrete applcaton. Concernng the stochastc epdemc models, the possblty of developng an exhaustve comparatve study s a promsng research topc for some forthcomng study. At present, we may comment that Padé approach used by MATLAB s prmarly concerned wth small dense matrces and large sparse matrces (as the trdagonal matrx of the SIS model). In these cases, the MATLAB routnes are perhaps more straghtforward than the numercal nverson methods. However, the numercal nverson methods requre nether the computaton of the whole matrx nor ts egenvalues. Moreover, the recursve schemes developed n ths paper explot the specal transton structure of the epdemc models under study and avod subtractons, whch greatly mproves the stablty. Thus, we recommend our methods especally when the prmary attenton s put on accuracy rather than n computer tme requred, and as far as the populaton sze ncreases and the underlyng matrx s not su cently sparse (e.g. the SIR model and other more sophstcated epdemc models). On the other hand, we observe that the d erentaton of the Laplace transform equatons provdes the quckest way to obtan recursve schemes for computng the moments. In ths way, we avod the computaton of the powers of matrces nvolved n (R2). The above dscusson on the extncton tme needs to be completed wth a few comments regardng approxmatons and asymptotc results. A remarkable result establshes that the extncton tme, when the ntal dstrbuton equals the quas-statonary dstrbuton, follows a smple exponental dstrbuton. For a proof n the context of the nte brth-death process, we refer to Norden (982). Ths approxmaton has been extensvely used. For the generalzaton to bvarate epdemc models see, for nstance, Nåsell (999, 2002). The lterature for asymptotc expansons s very rch; as an example, we menton the papers by Doerng et al. (2005) and Nåsell (200). Whle many unvarate epdemc models can be formulated as brth-death processes, the SIR models provde a natural framework to deal wth the bvarate 4

5 case. Daley and Gan (999) consder an SIR formulaton to model stochastcally a general epdemc. Then, the extncton tme s nvestgated by usng generatng functons and Laplace transforms, but the authors conclude that the obtaned soluton s algebracally formdable. For the specal case where the epdemc starts wth one nfectve and the populaton sze s very small, Gan (965) gves smpler results. Bllard and Zhao (993) nvestgate the transent analyss of an SIR model and, consequently, the extncton tme dstrbuton can be derved from that study. However, the numercal mplementaton of ther results s lkely to be cumbersome and t seems to nvolve unstable elements (.e., subtractons, alternatng sgns). Barbour (975) nvestgates the asymptotc behavor of the extncton tme as the populaton sze tends to n nty. The computaton of the expected extncton tme s underlyng n the numercal examples gven n ths paper. Our contrbuton to the analyss of the extncton tme s twofold. The rst one s the computaton of the whole dstrbuton of the extncton tme by applyng numercal Laplace nverson algorthms. Although mean and varance summarze the man statstcal propertes of a random varable, knowledge of the whole probablty dstrbuton s of nterest n ts own rght. In partcular, the probablty that the length of the outbreak exceeds a certan crtcal threshold may be helpful to take preventve actons. Moreover, the mere knowledge of the rst moments s somewhat deceptve, when the dstrbuton under study s not unmodal. Ths s the case of the nal sze dstrbuton and the extncton tme dstrbuton of the SIR model, whch are bmodal (see, for example, Baley (975) and Barbour (975)). At ths pont, we remark that the methodology developed n ths paper s also helpful to explore the shape of the densty functon by dentfyng ts modes and the behavor at the tme orgn 0: A second contrbuton concerns the moments of the condtonal extncton tmes for the SIR epdemc model. By explotng the specal transton structure of ths model, we get stable recursve schemes for computng transforms and moments. Along the paper, we propose smple recursve schemes that avod subtractons. In ths way, we crcumvent the above mentoned drawbacks nherent to the computaton of formulae n (R) and (R2). Fnally, we now turn our attenton to the tme to nfecton and to the recovery tme. As far as we know, these two epdemc descrptors have not been studed yet n the framework of stochastc bologcal models. Stone et al. (2008) de ne the probablty that an ndvdual becomes nfected. In Subsecton 2.2., we show how ths probablty s related to the tme to nfecton. We perform an exhaustve analyss of these descrptors both for the brth-death model and the stochastc SIR model. In ths paper, we focus on the brth-death model and the basc formulaton of the stochastc SIR model. However, our results can be extended to other stochastc epdemc models. In any subsequent study, we would lke to extend our analyss to models wth kllng and catastrophes (see e.g. the papers by Coolen-Schrjner and van Doorn (2006) and Artalejo et al. (2007)) and also to more complcated varants of the SIS and SIR epdemc models (see e.g. some recent publcatons n ths journal as the paper by McCormack and Allen (2007), 5

6 Clémençon et al. (2008) and Martns et al. (2009)). The rest of the paper s organzed as follows. In Secton 2, we ntroduce the stochastc brth-death model and the random varables representng the extncton tme, the tme to nfecton and the recovery tme. We develop algorthmc schemes for ther analyss, rst focusng on ther Laplace transforms and then proceedng to the study of the moments. Secton 3, presents a parallel analyss for the stochastc SIR model. In Secton 4, we present some selected numercal results. In partcular, n Subsecton 4. we employ the descrptors under study n the context of the head lce nfecton reported by Stone et al. (2008). The behavor characterstcs of the SIR model are numercally nvestgated n Subsecton 4.2. Fnally, Subsecton 4.3 contans some numercal results regardng the number of modes for the characterstcs of the SIS model. 2 Brth-death process We consder a closed populaton of N ndvduals, where each ndvdual s class- ed as ether a susceptble or an nfectve. Indvduals move from the susceptble to the nfected group and then they recover returnng to the susceptble pool. The stochastc model descrbng the evoluton of the epdemc can be seen as a brth-death process fi(t); t 0g wth state space S = f0; : : : ; Ng, where I(t) gves the number of nfectves at tme t. The brth rates, correspondng to nfectons, are denoted by and the death rates, correspondng to recuperatons, are denoted by, = 0; : : : ; N. The nfectons are supposed to occur because of a contagous dsease. Hence, when there are no nfectves, the process stays there forever. The other states are assumed transent. More spec cally we assume that 0 = N = 0 = 0, whle ; : : : ; N and ; : : : ; N are strctly postve. A partcular example of brth-death processes s, for nstance, the classcal SIS model, wth = (N )=N for the brth rate and = for the death rate, where s the contact rate and s the recovery rate per ndvdual. A more general model s the Verhulst model (see Nåsell (200) for detals), wth nfecton rates = ( ( =N)) ; for 0 N, N = 0, and recovery rates = ( + ( 2 =N)), for 0 N: 2. Extncton tme Let us assume that at the ntal tme t = 0 the populaton has nfectve ndvduals, and de ne a contnuous random varable L to be the extncton tme of the epdemc gven the current populaton state. Ths varable can be seen as the absorpton tme by the state 0 gven that I(0) = : Next we ntroduce some notaton for absorpton probabltes, Laplace transforms and moments of L, for 0 N. Let us de ne u = P fl < g; 0 N; ' (s) = E[e sl ]; 0 N; Re(s) 0; M k = E[L k ]; 0 N; k 0: 6

7 Frst we observe that u = ; for N; because the set f; :::; Ng s a non-decomposable set of states. Theorem provdes a computatonally stable recursve scheme, from whch the computaton of Laplace transforms can be done at a low computatonal cost. More concretely, the proposed scheme only deals wth algebrac operatons nvolvng postve terms, whch guarantees stablty even for large values of N. Once the Laplace transforms ' (s) have been computed, the densty functons f L (x) (or, alternatvely, the probabltes P fl > xg) can be obtaned numercally by usng Fourer seres methods. Theorem The Laplace transforms ' (s), for N; are computed by the equatons ' N (s) = ' (s) = N D N s N + (s + N )(s + g N ) ; () NX D k ky N n Y k + ' k s + g k= n= n + N (s) ; N ; n s + g k + k k= (2) where the coe cents g and D, for N scheme ; are gven by the recursve g = ; (3) s + g g = ; 2 N ; (4) s + g + D = ; (5) D = D ; 2 N : (6) s + g + Proof. Condtonng on the exponentally dstrbuted tme to the rst transton, we have ' 0 (s) = ; ' (s) = ' s + + (s) + ' s (s); N: (7) The above equaton (7), for N ; can be expressed as follows ' (s) + ' (s) + ' + (s) = ; (8) where = 0; = ; 2 N ; = s + + ; N ; = ; N ; = ; = 0; 2 N : 7

8 Usng a forward-elmnaton-backward-substtuton procedure, the tr-dagonal system (8) becomes where G ' (s) + ' + (s) = D ; N ; (9) G = = s + + ; G = = s + + G ; 2 N ; G D = = ; D = D G = D G ; 2 N : In order to avod negatve terms, we ntroduce a new set of coe cents g = G (s + ), for N. Then, we easly nd that the coe cents g and D are as clamed n (3)-(6). Now we use equaton (9) and express ' (s) n terms of ' + (s) ' (s) = D ' + (s) G = D + ' + (s) s + g + ; N : (0) Iteratng (0) we obtan equaton (2). Now usng (7) for = N, we have (s + N )' N (s) = N ' N (s): () Fnally, from equaton (2), for = N, and () we get equaton (). On the other hand, the value of the densty functon at the pont x = 0 follows by d erentatng expresson (R) whch yelds ; f = ; f L (0) = 0; otherwse. Next we focus on the calculaton of the moments M k ; 0 N, for any arbtrary non negatve nteger k. Note that moments of order k = 0 are M 0 = u =, for 0 N. For k, by d erentatng equaton (7) k tmes wth respect to s and settng s = 0; we nd that M k 0 = 0; (2) ( + )M k = M k + M k + + km k ; N: (3) These equatons were also derved by Norden (982) usng the backward equatons and also, for the partcular case of an SIS epdemc wth external source of nfecton, by Stone et al. (2008), from the moment generatng functon. 8

9 Moments of L, for N, are determned recursvely on k by usng the followng formula, whch agrees wth equaton (6.8) n Norden (982) and generalzes the correspondng one stated n Stone et al. (2008): where 0 = and j = X M k = j Q m= N X j j=0 n=j+ m m, for j N : 2.2 Tme to nfecton and recovery tme km k n n n ; (4) Let us consder the populaton at an arbtrary tme t and suppose that there are nfected ndvduals at ths tme, for 0 N. We mark one of the N non nfected ndvduals and denote by S a random varable representng the tme untl the selected ndvdual gets nfected. Obvously, S 0 = + and the case = N has no sense. To study the varables S, for 0 N, we de ne v = P fs < g; 0 N ; (s) = E[e ss fs<g]; 0 N ; Re(s) 0; fm k = E[S k fs<g]; 0 N ; k 0; where, for an event A; A s the ndcator random varable that takes the value when the event A occurs and s 0 otherwse. Note that the probabltes v are strctly between 0 and, for N. Indeed, P fs < g + : : : N N + > 0 and P fs = g N + > 0, hence P fs < g <. + : : : A rst step argument, condtonng on the dentty of the next nfected ndvdual (.e., ether a recovery, an nfecton for a non-tagged ndvdual or the nfecton of the marked ndvdual), shows that the Laplace transforms, (s), satsfy the followng set of equatons: 0(s) = 0; (5) (s) = N s + + (s) + s + + N +(s) + ; N : (6) s + + N For every s; the system of equatons (5) and (6) s tr-dagonal and the coe cent matrx s strctly dagonally domnant. We agan solve the system by usng a forward-elmnaton-backward-substtuton method. After some algebra, we obtan a stable recursve scheme whch appears n the followng theorem. 9

10 Theorem 2 The Laplace transforms (s), for N ; are computed by the equatons 2 N (s + g N 2 + N 2 ) + N (s) = N D N 2 2(s + N )(s + g N 2 + N 2 ) + N (2(s + g N 2 ) + N 2 ) ; (s) = NX 2 k= D k N k k N k NY 2 + N (s) k= ky n N n s + g n= n + n N n k N k s + g k + k N k where the coe cents g and D, for N scheme (7) ; N 2; (8) 2; are gven by the recursve g = ; (9) s + g + N + g = ; 2 N 2; (20) s + g + D = D = N ; (2) N + D ; 2 N 2: (22) s + g + Proof. The proof follows along the lnes descrbed n the proof of Theorem, so t s omtted. The startng value of the densty functon f S (x) follows from the Tauberan result f S (0) = lm s! s (s). In the lght of (6), t gves f S (0) = N ; for N : Observe that v = M f 0 = P fs < g = (0). Consequently, the probabltes v ; 0 N, can be determned by settng s = 0 n (7)-(22). It should be ponted out that the resultng equatons for the probabltes v correspond to those gven by Stone et al. (2008) for the probablty that an ndvdual becomes nfected. n o We now concentrate on the calculaton of the moments fm k ; 0 N. By d erentatng equatons (5) and (6) k tmes, and evaluatng at s = 0, we nd that fm k 0 = 0; ( + ) M f k = M fk N + fm + k + km N fk ; N : (23) Ths system of equatons provdes, after some algebra, a stable recursve scheme for the computaton of M f k, for N and k, havng a smlar 0

11 structure as the one appearng n Theorem 2: In fact, for each xed k, the followng smple mod catons are needed: () replace (s) by M f k, for N ; () set s = 0 n equatons (7)-(20), and () replace D n equatons (2) and (22) by D = km f k and D = km fk + D (g + ) ; for 2 N 2: Now, each teraton allows us to compute the unknown moments of order k n terms of the moments of one order less. Note that the moments of order k = 0 are v. Note, however, that the system (23) cannot lead to a more explct expresson as n the case of equaton (3). The reason s that the recurson scheme (23) s of second-order wth no constant coe cents and, more mportantly, the sum of the coe cents at each row of the system s not 0, as n the case of equatons (3). Therefore, t s not possble to reduce the order of the scheme by consderng the d erences. The same holds as well for the other recursve schemes for the computaton of the moments of varous descrptors that we report n the rest of the paper (see e.g. equatons (29) and (34)). Next we study the recovery tme, R ; de ned as the tme untl a tagged nfected ndvdual gets recovered, gven that the populaton conssts of nfectve ndvduals, for N. Obvously, R 0 has no sense. To study these random varables we de ne w = P fr < g; N; (s) = E[e sr ]; N; Re(s) 0; m k = E[R k ]; N; k 0: It s clear that the recovery tme of a marked ndvdual s at most the tme to extncton of the epdemc. Thus, we have that R L, for all N. Hence, as t was shown n Secton 2., from u = P fl < g = we conclude that w = m 0 = P fr < g =, for N. Agan, a rst step argument gves that the Laplace transforms, (s), satsfy (s) = s + + (s) + s (s) + ; N; (24) s + + and the ntal densty value s f R (0) = lm s! s (s) = ; for N: For k, by d erentatng equaton (24) k tmes wth regard to s and settng s = 0, we get that the moments of order k satsfy the equatons ( + )m k = m k + m k + + km k ; N: Once more, we notce that the coe cent matrx n (24) s strctly dagonally domnant. Ths guarantees the exstence of nte solutons both for transforms and moments. In addton, t s possble to develop algorthmc schemes for

12 ther computatonal analyss, whose structure s smlar to the one appearng n Theorems and 2. Remark 3 When the rate at whch recovery events occur s proportonal to the number of nfected ndvduals (.e., the case = ), we observe mmedately that the recovery tmes, R, for N, are exponentally dstrbuted wth rate ; ndependently of the number of nfectve ndvduals n the populaton. We may consder uncondtonal versons of the tme to nfecton and the recovery tme. Let us denote such versons as S and R, respectvely, and ther correspondng Laplace transforms as (s) and (s), for Re(s) 0: In a general settng, we may consder (s) = (s) = NX = a (s); NX b (s); = where the postve weghts a and b satsfy that P N = a = P N = b = : There exst varous ways to choose approprate weghts. A reasonable possblty s to employ the quas-statonary probabltes, q, of havng nfectve ndvduals gven that the extncton has not occurred yet. 3 Stochastc SIR model Let us consder agan a closed populaton of N ndvduals, subdvded nto susceptble, nfectve and removed or mmune ndvduals. A susceptble ndvdual can get nfected and then can recover. Those recovered ndvduals are assumed permanently mmune to further nfectons. As a result, we refer to them as removed ndvduals, whle n fact they reman n the populaton. The denomnator N n the transton rates (see next paragraph) shows that the contrbuton of the mmunes s to waste potental nfecton capacty from the nfectves. The assumpton that the total populaton s closed mples that, at any tme t; S(t) + I(t) + R(t) = N, where S(t), I(t) and R(t) denote the number of susceptbles, nfectves and removals, respectvely, at tme t: The populaton dynamcs s descrbed n terms of a bdmensonal contnuous tme Markov chan f(i(t); S(t)); t 0g: Transtons from a state (; j) can be ether to state ( + ; j ) at a rate j ; due to an nfecton, or to state ( ; j) at a rate ; due to a removal. We consder that the epdemcs conssts of a sngle outbreak so the transton from (0; j) to (; j ) s not permtted. In addton, we have some trval null rates, that s 0 = 0 = 0: Usually, the transton rates are assumed to have the form j = j=n and =, where the parameters > 0; > 0 are dent ed as the contact rate and removal rate, respectvely. Varous authors suggest that the model mght be more realstc by allowng contact and removal 2

13 rates to depend upon the number of susceptbles and nfectves present n the populaton. In that sense Severo (969) and more recently Clancy and Green (2007) suggest potental functons. S(t) R 3 R 23 R R 2 R 22 R 32 R R R 2 R 3 R 4 R I(t) Fgure. States and transtons of the SIR epdemc model In what follows, let us assume that the populaton ntally conssts of m nfectves and n susceptbles. The state space of the SIR epdemc model s S = f(; j); 0 j n; 0 m + n jg, wth absorbng states f(0; j); 0 j ng. The rest of states are transent and after leavng one of them, the chan never returns to ths state. If we take j as the man level and arrange the states n lexcographc order, then we observe that the generator Q has a trangular structure whch s really helpful to get stable recursve algorthms. Fgure, shows the state space and the transtons rates when (m; n) = (3; 3): 3. Extncton tme We ntroduce the random varable L j as the extncton tme of the outbreak gven that the current state s (; j). Let us de ne the absorpton probabltes u j = P fl j < g: We notce that u j = for all transent states. Ths s due to the fact that the submatrx of Q governng the moton n the set of transent states s nvertble because t s a trangular matrx wth non-null dagonal elements. Next we de ne the Laplace transforms and moments of L j : ' j (s) = E[e slj ]; (; j) 2 S; Re(s) 0; M k j = E[L k j]; (; j) 2 S; k 0: Usng a rst step argument we get that the functons ' j (s); (; j) 2 S; satsfy 3

14 ' 0j (s) = ; 0 j n; (25) ' j (s) = j ' s + j + ;j (s) + ' s + j + +;j (s); (26) Solvng equaton (26) for j = 0, we have ' 0 (s) = s + ' ;0 (s) = = 0 j n ; m + n j: Y k= k s + k ; m + n: (27) Note that the equatons (25)-(27) can be combned, n the natural order j n and m + n j, to determne recursvely the values of ' j (s); at a xed pont s. In partcular, we can get the Laplace transform of the extncton tme from the ntal state (m; n). After that, the use of numercal nverson algorthms permts to determne numercally P fl mn > xg, that s the probablty that an outbreak, startng wth m nfectves and n susceptbles, wll last more than x tme unts. We observe that the densty functon at x = 0 s gven by f Lj (0) =, f = ; and t s 0 otherwse. For determnng the moments of L j of any arbtrary order k 0; we rst notce that M k 0j = 0; 0 j n; k 0; and for the transent states M 0 j = u j = ; 0 j n; m + n j: In order to get the moment of order k, by d erentatng equatons (26) and (27) k tmes wth respect to s and settng s = 0, we nd that M k 0 = k M k j = X l= M k l0 l ; m + n; k ; (28) M k ;j + j M k k +;j + M k j ; (29) j + j + j + j n; m + n j; k : Formulae (28) and (29) provde an e cent recursve scheme for the computaton of Mj k n the natural order. 3.2 Tme to nfecton and removal tme Let us assume that the ntal state of the populaton s (I(0); S(0)) = (m; n), wth m : We choose one of the susceptble ndvduals, we mark t and nvestgate the random varable S mn de ned as the sojourn tme untl the marked ndvdual becomes nfected. We note that t s possble that the outbreak ends before the nfecton of the marked ndvdual and consequently P fs mn = g > 0: 4

15 In order to nvestgate the probablstc behavor of S mn we ntroduce, for the rest of possble states, analogous random varables, S j ; de ned as the tme to nfecton of the marked susceptble ndvdual condtoned to the current state (; j) 2 S: Obvously, S 0j = +; for j n; and, for 0 m+n; S 0 have no sense because there are not susceptble ndvduals. The rest of probabltes v j = P fs j < g, assocated to the transent states, are strctly between 0 and : Indeed, for j n and m + n j; we notce that P fs j < g j j+ ::: +j ; +j ;+ > 0 and P fs j = g +j j+ ::: j+ > 0: Next we nvestgate the Laplace transforms and moments of S j ; for j n and m + n j; so we de ne j(s) = E[e ssj fsj<g]; j n; 0 m + n j; Re(s) 0; fm k j = E[S k j fsj<g]; j n; 0 m + n j; k 0: By a rst step argument, condtonng on the next event (.e., ether a removal, an nfecton for a non-tagged ndvdual or the nfecton of the marked ndvdual) we nd that the Laplace transforms satsfy 0j(s) = 0; j n; (30) j(s) = j j s + j + ;j(s) + +;j s + j + j (s) j + ; j n; m + n j: (3) s + j + j We can solve the system of lnear equatons (30) and (3) wthout evaluatng ts nverse matrx. Notce rst that equaton (3), for j =, yelds (s) = s + + ; and ths explct expresson provdes the startng pont to get the rest of Laplace transforms. Va a stable procedure, we compute recursvely j (s) n the natural order j n and m + n j: Moreover, we notce that f Sj (0) = j j ; for j n and m + n j: Now we observe that v j = M f j 0 = P fs j < g = j (0): Thus, t follows that the probabltes v j satsfy the equatons (30) and (3), wth s = 0. We can also get the moments of order k, M fk j ; as the soluton of the followng system of lnear equatons: fm k 0j = 0; j n; k ; fm k j = M fk j + ;j + j j fm +;j k j + j k + M fk j ; j n; m + n j; k : j + 5

16 Fnally we study the removal tme, R j ; de ned as the tme untl a tagged nfected ndvdual gets mmunty, gven that the current populaton has nfectve and j susceptble ndvduals, for 0 j n and m + n j. In fact, we want to compute the characterstcs of the random varable R mn but ths computaton nvolves all the varables R j : Frst we notce that for any state (; j), the removal tme s always shorter than the extncton tme of the outbreak, that s R j L j. Hence, f we de ne the probabltes w j = P fr j < g we can conclude that w j = ; for 0 j n and m + n j: Let us ntroduce some notaton for the Laplace transforms and moments j (s) = E[e srj ]; 0 j n; m + n j; Re(s) 0; m k j = E[R k j]; 0 j n; m + n j; k 0: We now observe that the Laplace transforms satsfy j (s) = s + j + ;j (s) + j s + j + +;j (s) + ; 0 j n; m + n j: (32) s + j + For any xed j, formula (32) provdes a recurson for computng the transforms j (s); for = ; : : : ; m + n j, n terms of +;j (s) whch has been computed n the prevous step. The startng densty value s now gven by f Rj (0) = ; for 0 j n; m + n j: D erentatng equaton (32) wth respect to s and settng s = 0; yelds m 0 j = w j = ; 0 j n; m + n j; (33) m k j = m k ;j + j m k +;j j + j + k + m k j ; 0 j n; m + n j + j; k : (34) Then, we can compute m k j recursvely from equatons (33) and (34), n the order k 0; 0 j n and m + n j. Remark 4 In agreement wth Remark 3, n the case = we observe that the removal tmes R j, for 0 j n and m + n j; are exponentally dstrbuted wth rate, ndependently of the number of susceptble and nfectve ndvduals n the populaton. 6

17 4 Numercal examples In ths secton, we present some numercal llustratons of the theoretcal results. A rst set of numercal experments concerns the applcaton to outbreaks of head lce, Pedculus humans capts. To ths end, n Subsecton 4., we employ the data set provded by Stone et al. (2008) correspondng to a study carred out n UK schools. The underlyng epdemc model s a stochastc SIS model wth an external source of nfecton. In Subsecton 4.2, we nvestgate the n uence of varous system parameters n the behavor characterstcs of the SIR epdemc model. Some specal attenton s pad on the dent caton of the modes. In fact, the analyss of the modes s extended n Subsecton 4.3 to the SIS model. 4. An applcaton to head lce nfectons Head lce, or pedculoss capts, s an mportant health problem among chldren worldwde. Although head lce rarely cause drect harm, t can cause dstress to many patents. Health care professonals, school admnstratons and parents seek frequently for solutons. Snce head lce are extremely contagous, t s mportant to observe promptly the outbreak and to reduce the transmsson rate. Some steps that can be followed to prevent the spread nclude regular checks n the schools, exstence of detecton campagns, as well as the engagement of parents. Once the outbreak starts, t s mportant to avod head-to-head contact, not to share combs, towels and other personal hygene objects. The common treatment optons are envronmental decontamnaton, mechancal removal and use of topcal nsectcdes (lndane, malathon). For further detals, we refer the reader to the recent studes by Damants et al. (2009) and Ibarra et al. (2009), and also to the references theren. In what follows, we deal wth the SIS epdemc model wth an external source of nfecton studed by Stone et al. (2008). Ths means, that we have a closed populaton of N ndvduals and assume that nfecton and removal rates are respectvely = (N )( + =N) and =, for 0 N, where denotes the external rate of nfecton. Snce > 0; we observe that 0 > 0 and the resultng brth-death process has no absorbng states. At ths pont, we stress that all our results n prevous Secton 2 reman vald, when we focus on the dynamcs of the nfecton durng an ndvdual outbreak. In other words, n ths model extncton means the rst moment of no nfectves. On the other hand, we notce that an outbreak starts when there s an external nfecton; that s, = : However, the outbreak can be detected later on (or ts control could start after some tme), so n our numercal experments we consder several choces for : To llustrate how our results can be used to nvestgate propertes of the nfecton spread, we next assume the model parameters consdered by Stone et al. (2008). They refer to the data from 3 Welsh (UK) schools. For a populaton sze of N = 00 and = (.e., one ntal nfectve), they set at.0 (.e., the unt tme s the expected tme of the nfectous perod) and estmate the rates = :02 and = 0:0 by the maxmum lkelhood method. These system 7

18 parameters provde our basc scenaro along ths subsecton. Frst, we focus on the cumulatve dstrbuton functon of the extncton tme, F L (x) = P fl xg. The computaton of F L (x) has been done by usng numercal nverson methods (see Secton ), nvolvng the Laplace transforms equatons shown n Theorem. In Fgure 2, we dsplay three curves correspondng to I(0) = = ; 25 and = = 2 = Fgure 2. Dstrbuton functon of L The dstrbuton of L presents heavy tals when there are larger number of ntal nfectves n the populaton. It seems ntutvely plausble that the duraton of an outbreak s stochastcally larger for ncreasng, havng xed. In other words, for < j, one expects that L st L j, where the symbol st denotes the usual stochastc order relaton wth respect to the dstrbuton functon. Ths result can be rgorously establshed usng the followng couplng argument. For < j, let I (t) and I j (t) be ndependent realzatons of the number of nfectves startng from and j nfectves, respectvely. Then, we de ne ei j (t) = Ij (t); f t T; I (t); f t > T; where T = nf ft 0 j I (t) = I j (t)g : We notce that I j (t) could be equal to I (t) + at some tme t < T, but the probablty that I (t) has a postve jump and I j (t) has a smultaneous negatve jump s zero. Ths elmnates the possblty that I j (t 0 ) < I (t), for any t 0 2 (t; T ). Moreover, the strong Markov property guarantees that the 8

19 constructed process I e j (t) s a Markov process, stochastcally equvalent to I j (t), and by constructon sats es that I e j (t) I (t): Ths proves the desred result. The nterested reader can found a detaled descrpton of the couplng method n Thorsson (2000). For nce applcatons to epdemcs see the book by Andersson and Brtton (2000). In addton, Fgure 2 shows the mportance of detectng the epdemc process at an early stage (.e., when t nvolves a few nfectve ndvduals). Note that 78% of the outbreaks startng wth only one nfectve ends before 20 unts tme. Whle, when the nfectous dsease a ects ntally 25 ndvduals, the tme L 25 s longer than 20 tme unts n more than 67% of the nfectons. Ths proporton ncreases to 7% when = 50: Now we show how the dstrbuton functon can be used to halve the mean length of an ndvdual outbreak wth a gven accuracy. We observe that E[L ] = 3:74; so E[L ]=2 = 6:8705: The objectve s to see P fl 6:8705g as a functon of, let us say f(), n order to determne the threshold at whch we get that f() > 0:8: The numercal evaluaton shows a monotone behavor on and t readly yelds f(0:70) = 0:8009 and f(0:7) = 0:7968, so we have = 0:70: Thus, the concluson s that the adopton of polces allowng to drop down the contact rate n = :02 0:70 = 0:32 unts would mply that the length of the outbreak s halved wth a probablty greater than 0.8. The numercal results suggest that the duraton of an outbreak s stochastcally ncreasng wth respect to. Once more, a rgorous proof of ths fact requres an appeal to couplng methods. Gven the ntal state, we observe that the next transton of the process fi(t); t 0g s determned by a competton between two exponentally dstrbuted random varables, let us say Exp( ) and Exp( ), wth rates and. For 0 >, let I (t) and I 0 (t) be two ndependent realzatons assocated wth the contact rates and 0, respectvely. Both realzatons start wth nfectves at tme t = 0. In what follows, the prme notaton wll be used to denote other characterstcs assocated wth the SIS model wth contact rate 0. We now observe that the exponental varable Exp( 0 ) can be expressed as the mnmum between two ndependent exponental varables, let us say Exp( ) and Exp(! ), wth! = ( 0 )(N )=N. To construct I (t) and an equvalent realzaton I e 0(t) of I0 (t); we use common varables Exp( ) and Exp( ): Let N be the number of ncdents of nfecton durng an outbreak wth ntal nfectves. For j N ; de ne T j to be the tme at whch the jth nfecton occurs n the SIS model wth contact rate. Note that T =, f N = 0. Smlarly de ne Tj 0 for the SIS model wth 0 : The couplng ensures that T 0 T. In fact, t follows that the updated number of nfectves at tme T 0 are gven by where : I (T 0 ) = ; f T 0 < T ; + ; f T 0 = T ; I 0 (T 0 ) = + ; 9

20 If T 0 = T, then the realzatons of the coupled processes contnue ther course. In contrast, f T 0 < T then we stop the couplng mechansm, update the ntal number of nfectves to ther current values and + ; and de ne bt = nf t > T 0 I (t) = I 0 + (t). It may occur that the outbreak of the SIS model wth rate ends before T b (.e., L < T b ), showng that L < L 0 + ; otherwse, t follows that I ( T b ) = I 0 + ( T b ) and the above constructon must be restarted. The desred result follows, snce the outbreak ends n a nte tme wth probablty one. P fs < g = 0:05 = 0:5 = :0 = 5:0 = 0: = 0: = : = 2: = 5: Table. Probablty of becomng nfected Next we present results for S, the tme tll the nfecton of a selected nonnfected ndvdual, gven that there are nfectves n the populaton. In Table, we dsplay results for the probablty that the selected ndvdual becomes nfected before the end of the outbreak. We stll keep N = 00 and = 0:0 but we allow the contact and the recovery rates to vary as 2 f0:05; 0:5; :0; 5:0; 0:0g and 2 f0:5; :0; 2:0; 5:0g. For any xed par (; ), the cell gves from top to bottom these probabltes, when the ntal number of nfectves vares as = ; 40 and 90; respectvely. We observe that P fs < g ncreases as a functon of : Accordng to the ntuton, the probablty of beng nfected shows an ncreasng behavor for ncreasng contact rates and decreases for ncreasng recovery rates. In Fgure 3, we come back to the basc scenaro (.e., we set (; ; ) = (:0; :02; 0:0)). Then, we draw the dstrbuton functon of the tme to nfecton, restrcted to ndvduals who become nfected before the outbreak ends; that s P fs x js < g. The graphs correspond to, 25 and 50 ntal nfectves. If we start wth nfectve, the selected ndvdual becomes nfected n at most 5 unts tme wth probablty 0.39, but ths probablty s around 0.70 when the nfectous dsease nvolves at least 25 ndvduals. In general, for a xed tme x, we can see that the probablty P fs x js < g ncreases as a functon of the ntal number of nfectves. Ths shows agan the mportance of an early detecton of the outbreak. 20

21 Snce the probablty g() = P fs < g s an ncreasng functon of, we may nd the value e such that the probablty of becomng nfected s su cently small; that s g() < : For example, for = 0:03 we observe that g(0:25) = 0:0297 and g(0:26) = 0:0304, so e = 0:25: Ths llustrates that the computaton of P fs < g helps to understand how much the contact rate should be reduced n order to control the epdemc spread at a desred level = 5 = 25 = Fgure 3. Dstrbuton functon P fs x js < g The results from Welsh schools data were summarzed by Stone et al. (2008) wth a school of sze 00 and one ntal nfectve. However, the data collected n Table 3 of that paper shows that the number of pupls checked n each school vares n a range from 24 (mnmum sze) to 20 (maxmum sze). Moreover, n ther dscusson, Stone et al. comment that t s mpossble to know when an outbreak starts and ends, unless frequent checks of chldren are performed. These practcal d cultes provde some motvaton to study the epdemc characterstcs for varous choces of N and : Ths s done n the followng. Table 2 llustrates the behavor of the two rst moments of L (.e., the expectaton and the standard devaton) and the probablty of becomng nfected, when N vares n a range closer to the range observed n Welsh schools. The ntal number of nfectves takes values = ; d0:05ne ; d0:0ne ; d0:25ne and d0:40ne ; where dxe s the celng functon that gves the smallest nteger larger or equal than x. The entres n the table measure the ncreasng e ect of the populaton sze and the ntal number of nfectves on the three selected nfecton characterstcs. Moreover, for a xed N, we observe that the three characterstcs exhbt a wde range of varaton. For example, for N = 00; the probablty of nfecton P fs < g vares from to Ths shows agan the mportance of detectng the epdemc soon n order to facltate 2

22 the e ort requred to control the nfecton spread. The noteworthy d erences observed for d erent values of N motvate the need of an accurate estmaton of the populaton sze of each school. If the objectve s to gan nsght for the whole Welsh area, then a mxture, wth approprate weghts, of the results obtaned for several choces of N could be helpful. E[L ] (L ) N = 30 N = 65 N = 00 N = 35 N = 70 N = 205 P fs < g = = d0:05n e = d0:0n e = d0:25n e = d0:40n e Table 2. Performance measures varyng N and 4.2 SIR epdemc model In ths subsecton, we consder an SIR epdemc model wth N = 30: In Tables 3 and 4, we respectvely gve the mean and the standard devaton of the extncton tme L mn for d erent choces of and : From top to bottom each cell contans the entres for the ntal states (m; n) 2 f(; 29); (5; 5); (29; )g: E[L mn ] = 0:05 = 0:5 = :0 = 5:0 = 0: = 0: = : = 2: = 5: Table 3. The mean extncton tme E[L mn ] 22

23 Frst of all, t should be ponted out that the classcal SIR model can be non-dmensonalsed (see Murray (2002)). Once the tme t has been rescaled as = t, the only relevant parameter s the rato. Ths explans why the expectatons assocated wth the par (; ) = (:0; :0) halve the correspondng expectatons for (; ) = (0:5; 0:5), whle the quanttes for (; ) = (5:0; 5:0) are reduced tenfold. If we x and (m; n); then we conclude that E[L mn ] seems to have a maxmum as a functon of : We also pont out that t s easy to prove that E[L mn ]! P m k=0 k, as! 0; whle E[L mn]! P m+n k=0 k ; as! : On the other hand, t s clear that E[L mn ] decreases wth ncreasng values of : In fact, for xed choces of and (m; n); t s easy to prove that E[L mn ]!, as! 0; and E[L mn ]! 0, as! : Fnally, for a xed par (; ), we observe that among the values of (m; n) 2 f(; 29); (5; 5); (29; )g; E[L mn ] s maxmzed when (m; n) = (29; ) for < ; whle for the maxmum s reached at the balanced ntal state (5; 5): (L mn ) = 0:05 = 0:5 = :0 = 5:0 = 0: = 0: = : = 2: = 5: Table 4. The standard devaton (L mn ) The man conclusons nferred from Table 4 are that (L mn ) decreases as a functon of ; but t has a maxmum as a functon of : A plausble explanaton of ths behavor s as follows. If we vew as a scalng tme parameter (see also Table 3), then we understand that ncreasng reduces the mean and the varance of L mn : When = <, only small epdemcs are expected, so the varance s small. In the case where = s slghtly above one, we may nd small and bg epdemcs and, as a result, the varance ncreases. Fnally, for large values of =; a rapd extncton s unlkely, so the varance becomes small agan. For a xed par (; ); several d erent patterns can be observed n the table. 23

24 (β, γ) = (0.05, 0.5) (β, γ) = (0.0, 0.5) Fgure 4. Densty functon of L ;29 when = 0: (β, γ) = (0.05, 5.0) (β, γ) = (0.0, 5.0) Fgure 5. Densty functon of L ;29 when = 5:0 In Fgures 4 and 5, the epdemc s allowed to start wth a sngle nfectve. Then, the densty functon of the extncton tme L ;29 ; s plotted for the parameter choces (; ) 2 f(0:05; 0:5); (0:05; 5:0); (0:0; 0:5); (0:0; 5:0)g. Denstes 24

25 assocated wth pars where < present a sngle mode at x = 0, whle the case > leads to bmodal dstrbutons. In ths case, we observe a larger proper mode when the rato = s larger (.e., the case (; ) = (0:0; 0:5)). The ntal value f L;29 (0) s n agreement wth the Tauberan result gven n Subsecton 3.. Ths bmodal nature of the dstrbuton agrees wth the asymptotc approxmaton obtaned by Barbour (975); see Fg. 2 n that paper. Barbour (see pages n that paper) gave an ntutve nterpretaton to the bmodal behavor n terms of the trajectores assocated wth large epdemcs. Those trajectores spend tme twce near 0: The rst tme s due to the ntal condton m = ; whle the second tme s consequence of takng = su cently large, whch make most realzatons of the epdemc lead to a major outbreak, wth relatvely long extncton tme. A smlar dscusson of the bmodal nature of the nal sze dstrbuton s gven on page 00 n Baley (975) (see also Nåsell (995)). If, then one mght expect only a mnor epdemc, so the dstrbuton s unmodal. In contrast, f >, for n su cently large, then ether a mnor epdemc occurs or a major epdemc occurs, so the dstrbuton s bmodal. For a better understandng of the bmodal nature of the extncton tme densty, we have performed some addtonal numercal experments. For the ntal state (m; n) = (; 99) and = :0, we have found that the densty has two modes, one at the orgn 0 plus a proper postve mode, whle remans greater than b = :273: For, b only the extreme mode at 0 s observed. The fact that the crtcal value of the contact rate for the dstrbuton to be bmodal s larger than one (.e., b > ) s to be expected from the dscusson gven n Baley (975) and Nåsell (995) for the dstrbuton of the total sze of the epdemc. On the other hand, for xed = :0 and = :5, we have ncreased the ntal number of nfectves m, but keepng m + n = 00: The concluson was that f Lmn (x) has two postve modes for 2 m 5. However, for the par (m; n) = (6; 94); we observed only one postve mode. These results corroborate that the exstence of two modes s expected provded that m s small and the epdemc s well establshed. Fnally, n Table 5 we compute the rst two moments of the removal tme. In Remark 4, we have shown that the lnear case = leads trvally to the exponental dstrbuton. However, one way n whch the model can be extended s to consder state dependent potental rates j = a j b and = +c, as suggested by Severo (969) and Clancy and Green (2007). The powers a; b and c are representng the nfecton power, the safety power and the removal power, respectvely. In our numercal example, we assume that (m; n) = (5; 5) and j = 30p j, and = p, for 0 j 5; 0 30 j: For ths choce, as expected, we observe that R 5;5 does not follow the exponental law. We notce that both the expectaton, E[R 5;5 ]; and the standard devaton, (R 5;5 ); are ncreasng functons of but they are decreasng functons of : 25

26 E[R 5;5 ] (R 5;5 ) = 0:5 = :0 = 2:0 = 5:0 = 0:05 = 0:5 = :0 = 5:0 = 0:0 5:4676 4:3972 5: : :8965 4:6670 6:9986 5:7203 2:7036 2:7504 2: :988 2: :2473 2:9429 2:67727 :35027 :36204 :37520 :486 :02983 :0456 :06236 : : :5467 0: : :452 0:4397 0:4669 0:43885 Table 5. Mean and standard devaton of R 5;5 7:203 5: : :8560 :59909 : :5896 0: SIS model: Analyss of the modes The exstence of bmodal dstrbutons for the extncton tme n the SIR model gves an ntal motvaton for a parallel study n the SIS model. To ths end, we have carred out some numercal experments, not reported here, for populaton szes N 2 f50; 00g when the recovery rate s = :0 and the contact rate s chosen as 2 f:5; 2:0; 2:5g: In all cases, we have obtaned decreasng densty functons, f =, and unmodal denstes f > : To llustrate ths stuaton, n Fgure 6 we consder three denstes for a populaton of N = 00 ndvduals. As s to be expected, the value of f L (0) obtaned numercally s n agreement wth the theoretcal Tauberan result gven n Subsecton (, β, γ) = (,.5,.0) (, β, γ) = (,.5, 2.0) (, β, γ) = (2,.5,.0) Fgure 6. Densty functon of L when N = 00 In order to explan why the densty f L (x) s not bmodal, we recall that the extncton tme of the SIS model, when the epdemc perssts for a very long 26

27 tme, can be approxmated by an exponental dstrbuton wth a very small rate (see Nåsell (200) and Norden (982)). The at decreasng shape of such an exponental densty helps to explan why the overall densty of the extncton tme s not bmodal. In a second example, we deal wth the densty of the tme to nfecton, f S (x), for the case N = 00: The three curves plotted n Fgure 7 correspond to the contact and recovery rates and the ntal number of nfectves already used n Fgure 6. We now observe that the ntal value s f S (0) = =(N ) = =N. When > ; we see a proper mode whch agrees wth the fact that the epdemc s su cently large. In contrast, n the curve where < ; only the ntal mode at 0 s observed (, β, γ) = (,.5,.0) (, β, γ) = (,.5, 2.0) (, β, γ) = (2,.5,.0) Fgure 7. Densty functon of S when N = 00 As a general concluson, we menton that t s not trval to formulate ntutve explanatons regardng the number of modes and the shape of the dstrbuton. Ths problem depends on several facts ncludng the number of system parameters, the nature of the rates (e.g. lnear rates or logstc rates), the possble exstence of nteracton between d erent types of ndvduals (e.g. nfectves and susceptbles n the SIR model) and the reproducton factor. The theoretcal nvestgaton seems complcated. Thus, the Laplace transform methodology used n ths paper ams to provde a helpful tool. Acknowledgements The authors are grateful to the referees for ther constructve comments whch were certanly helpful to mprove the paper. We especally thank suggestons regardng the bmodal nature of the dstrbutons, computatonal ssues and 27

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