Chapter 21. Epidemics Diseases and the Networks that Transmit Them

Transcription

1 From the book Netorks, Crods, and Markets: Reasoning abot a Highly Connected World. By Daid Easley and Jon Kleinberg. Cambridge Uniersity Press, Complete preprint on-line at Chapter 21 Epidemics The stdy of epidemic disease has alays been a topic here biological isses mix ith social ones. When e talk abot epidemic disease, e ill be thinking of contagios diseases cased by biological pathogens things like inflenza, measles, and sexally transmitted diseases, hich spread from person to person. Epidemics can pass explosiely throgh a poplation, or they can persist oer long time periods at lo leels; they can experience sdden flare-ps or een ae-like cyclic patterns of increasing and decreasing prealence. In extreme cases, a single disease otbreak can hae a significant effect on a hole ciilization, as ith the epidemics started by the arrial of Eropeans in the Americas [130], or the otbreak of bbonic plage that killed 20% of the poplation of Erope oer a seen-year period in the 1300s [293] Diseases and the Netorks that Transmit Them The patterns by hich epidemics spread throgh grops of people is determined not jst by the properties of the pathogen carrying it inclding its contagiosness, the length of its infectios period, and its seerity bt also by netork strctres ithin the poplation it is affecting. The social netork ithin a poplation recording ho knos hom determines a lot abot ho the disease is likely to spread from one person to another. Bt more generally, the opportnities for a disease to spread are gien by a contact netork: there is a node for each person, and an edge if to people come into contact ith each other in a ay that makes it possible for the disease to spread from one to the other. This sggests that accrately modeling the nderlying netork is crcial to nderstanding the spread of an epidemic. This has led to research stdying ho trael patterns ithin a city [149, 295] or ia the orldide airline netork [119] cold affect the spread of a Draft ersion: Jne 10,

2 646 CHAPTER 21. EPIDEMICS fast-moing disease. Contact netorks are also important in nderstanding ho diseases spread throgh animal poplations ith researchers tracing ot the interactions ithin liestock poplations dring epidemics sch as the 2001 foot-and-moth otbreak in the United Kingdom [211] as ell as plant poplations, here the affected indiidals occpy fixed locations and diseases tend to hae a mch clearer spatial footprint [139]. And similar models hae been employed for stdying the spread of compter irses, ith malicios softare spreading beteen compters across an nderlying commnication netork [241]. The pathogen and the netork are closely intertined: een ithin the same poplation, the contact netorks for to different diseases can hae ery different strctres, depending on the diseases respectie modes of transmission. For a highly contagios disease, inoling airborne transmission based on coghs and sneezes, the contact netork ill inclde a hge nmber of links, inclding any pair of people ho sat together on a bs or an airplane. For a disease reqiring close contact, or a sexally transmitted disease, the contact netork ill be mch sparser, ith many feer pairs of people connected by links. Similar distinctions arise in stdying compter irses, here a piece of softare infecting compters across the Internet ill hae a mch broader contact netork than one that spreads by short-range ireless commnication beteen nearby mobile deices [251]. Connections to the Diffsion of Ideas and Behaiors. There are clear connections beteen epidemic disease and the diffsion of ideas throgh social netorks. Both diseases and ideas can spread from person to person, across similar kinds of netorks that connect people, and in this respect, they exhibit ery similar strctral mechanisms to the extent that the spread of ideas is often referred to as social contagion [85]. Haing considered the diffsion of ideas, innoations, and ne behaiors in Chapter 19, hy then are e reisiting this topic afresh in the context of diseases? In the context of or discssions here abot netorks, the biggest difference beteen biological and social contagion lies in the process by hich one person infects another. With social contagion, people are making decisions to adopt a ne idea or innoation, and or models in Chapter 19 ere focsed on relating the nderlying decision-making processes to the larger effects at the netork leel. With diseases, on the other hand, not only is there a lack of decision-making in the transmission of the disease from one person to another, bt the process is sfficiently complex and nobserable at the person-to-person leel that it is most sefl to model it as random. That is, e ill generally assme that hen to people are directly linked in the contact netork, and one of them has the disease, there is a gien probability that he or she ill pass it to the other. This se of randomness allos s to abstract aay qestions abot the mechanics of ho one person catches a disease from another for hich e hae no sefl simple models. This, then, ill be the concrete difference in or discssion of biological as opposed to

3 21.2. BRANCHING PROCESSES 647 social contagion not so mch the ne context as the ne classes of models, based on random processes in netorks, that ill be employed. In the next three sections, e discss some of the most basic probabilistic models for epidemics in netorks; e then consider ho these models proide insight into some basic qalitatie isses in the spread of disease, inclding synchronization, timing, and concrrency in transmission. Finally, e discss ho some of the models deeloped here are related to similar isses in genetic inheritance, here a kind of randomized propagation takes place throgh genealogical netorks. Before moing on to this, it is orth noting that randomized models can also sometimes be sefl in stdying social contagion, particlarly in cases here the nderlying decision processes of the indiidals are hard to model and hence more seflly abstracted as random. Often the to approaches decision-based and probabilistic prodce related reslts, and they can sometimes be sed in conjnction [62, 408]. Understanding the relationship beteen these methodologies at a deeper leel is an interesting direction for frther research Branching Processes We begin ith perhaps the simplest model of contagion, hich e refer to as a branching process. It orks as follos. (First ae.) Sppose that a person carrying a ne disease enters a poplation, and transmits it to each person he meets independently ith a probability of p. Frther, sppose that he meets k people hile he is contagios; let s call these k people the first ae of the epidemic. Based on the random transmission of the disease from the initial person, some of the people in the first ae may get infected ith the disease, hile others may not. (Second ae.) No, each person in the first ae goes ot into the poplation and meets k different people, reslting in a second ae of k k = k 2 people. Each infected person in the first ae passes the disease independently to each of the k second-ae people they meet, again independently ith probability p. (Sbseqent aes.) Frther aes are formed in the same ay, by haing each person in the crrent ae meet k ne people, passing the disease to each independently ith probability p. Ths the contact netork for this epidemic can be dran as in Figre 21.1(a) (ith k =3 and only the first three aes shon). We refer to sch a netork as a tree: it has a single node at the top called the root; eery node is connected to a set of nodes in the leel belo it; and eery node bt the root is also connected to a single node in the leel aboe it. The tree that forms the contact netork for the branching process is in fact infinite, since e contine defining aes indefinitely.

4 648 CHAPTER 21. EPIDEMICS (a) The contact netork for a branching process (b) With high contagion probability, the infection spreads idely (c) With lo contagion probability, the infection is likely to die ot qickly Figre 21.1: The branching process model is a simple frameork for reasoning abot the spread of an epidemic as one aries both the amont of contact among indiidals and the leel of contagion.

5 21.2. BRANCHING PROCESSES 649 No, hat is the behaior of an epidemic in this model? We can pictre the spread of the epidemic by highlighting the edges of the contact netork on hich the disease passes sccessflly from one person to another recall that each of these infections happens independently ith probability p. Ths, Figre 21.1(b) shos an aggressie epidemic that infects to people in the first ae, three in the second ae, fie in the third ae, and presmably more in ftre aes (not shon in the pictre). Figre 21.1(c), on the other hand, shos a mch milder epidemic (for a less contagios disease, ith a smaller ale of p): of the to people infected in the first ae, one doesn t infect anyone else, and the other infects only one frther person ho in trn doesn t pass it on. This disease has completely anished from the poplation after the second ae, haing infected only for people in total. The Basic Reprodctie Nmber and a Dichotomy for Branching Processes. Or last obseration abot Figre 21.1(c) reflects a fndamental property of branching processes: if the disease in a branching process eer reaches a ae here it fails to infect anyone, then it has died ot: since people in ftre aes can only catch the disease from others higher p in the tree, no one in any ftre ae ill be infected either. So there are really only to possibilities for a disease in the branching process model: it reaches a ae here it infects no one, ths dying ot after a finite nmber of steps; or it contines to infect people in eery ae, proceeding infinitely throgh the contact netork. And it trns ot that there is a simple condition to tell these to possibilities apart, based on a qantity called the basic reprodctie nmber of the disease. The basic reprodctie nmber, denoted R 0, is the expected nmber of ne cases of the disease cased by a single indiidal. Since in or model eeryone meets k ne people and infects each ith probability p, the basic reprodctie nmber here is gien by R 0 = pk. The otcome of the disease in a branching process model is determined by hether the basic reprodctie nmber is smaller or larger than 1. Claim: If R 0 < 1, then ith probability 1, the disease dies ot after a finite nmber of aes. If R 0 > 1, then ith probability greater than 0 the disease persists by infecting at least one person in each ae. We gie a proof of this claim in Section Een ithot the details of the proof, hoeer, e can see that the basic condition expressed in the claim comparing R 0 to 1 has a natral intitie basis. When R 0 < 1, the disease isn t able to replenish itself: each infected person prodces less than one ne case in expectation, and so een if it gros briefly de to the otcome of random flctations the size of the otbreak is constantly trending donard. When R 0 > 1, on the other hand, the size of otbreak is constantly trending pard. Notice, hoeer, that een hen R 0 > 1, the conclsion is simply that the disease persists ith positie probability, not ith absolte certainty: heneer p<1, then there

6 650 CHAPTER 21. EPIDEMICS is alays some chance that none of the first fe infected people ill scceed in infecting anyone else, casing the disease to die ot. In other ords, een an ltra-contagios disease can simply get nlcky and anish from the poplation before it has a chance to really get going. The dichotomy expressed by this condition has an interesting knife-edge qality to it hen R 0 is close to 1. In particlar, sppose e hae a branching process here R 0 is ery slightly belo 1, and e increase the contagion probability p by a little bit; the reslt cold psh R 0 aboe 1, sddenly reslting in a positie probability of an enormos otbreak. The same effect can happen in the reerse direction as ell, here slightly redcing the contagiosness of a disease to psh R 0 belo 1 can eliminate the risk of a large epidemic. And since R 0 is the prodct of p and k, small changes in the nmber of people k that each person comes into contact ith can also hae a large effect hen R 0 is near 1. All this sggests that arond the critical ale R 0 = 1, it can be orth inesting large amonts of effort een to prodce small shifts in the basic reprodctie nmber. Since R 0 is the prodct of the to terms p and k, it is in fact easy to interpret to basic kinds of pblic-health measres in terms of redctions to R 0 : qarantining people, hich redces the qantity k, and encoraging behaioral measres sch as better sanitary practices to redce the spread of germs, hich redces the qantity p. The branching process model is clearly a ery simplified model of disease-spreading; the strctre of the contact netork, ith no triangles at all, is reminiscent of or first pass at a model for the small-orld phenomenon in Chapter 20. Ths, in the next fe sections, e ill look at models that can handle more complex contact netorks. For these models, a dichotomy as simple as the one in the Claim aboe does not hold. Hoeer, the notion of the basic reprodctie nmber is still a sefl heristic gide to the behaior of more complex models; een hen epidemiological modelers do not hae a precise condition goerning hen an epidemic ill persist and hen it ill die ot, they find the reprodctie nmber R 0 to be a sefl approximate indication of the spreading poer of the disease The SIR Epidemic Model We no deelop an epidemic model that can be applied to any netork strctre. To do this, e presere the basic ingredients of the branching process model at the leel of indiidal nodes, bt make the contact strctre mch more general. An indiidal node in the branching process model goes throgh three potential stages dring the corse of the epidemic: Ssceptible: Before the node has caght the disease, it is ssceptible to infection from its neighbors.

7 21.3. THE SIR EPIDEMIC MODEL 651 Infectios: Once the node has caght the disease, it is infectios and has some probability of infecting each of its ssceptible neighbors. Remoed: After a particlar node has experienced the fll infectios period, this node is remoed from consideration, since it no longer poses a threat of ftre infection. Using this three-stage life cycle for the disease at each node, e no define a model for epidemics on netorks. We are gien a directed graph representing the contact netork; so an edge pointing from to in the graph means that if becomes infected at some point, the disease has the potential to spread directly to. To represent a symmetric contact beteen people, here either has the potential to directly infect the other, e can pt in directed edges pointing each ay: both from to and also from to. Since contacts beteen people are often symmetric, it is fine to se netorks here most edges appear in each direction, bt it is sometimes conenient to be able to express asymmetric contacts as ell. No, each node has the potential to go throgh the Ssceptible-Infectios-Remoed cycle, here e abbreiate these three states as S, I, and R. The progress of the epidemic is controlled by the contact netork strctre and by to additional qantities: p (the probability of contagion) and t I (the length of the infection). Initially, some nodes are in the I state and all others are in the S state. Each node that enters the I state remains infectios for a fixed nmber of steps t I. Dring each of these t I steps, has a probability p of passing the disease to each of its ssceptible neighbors. After t I steps, node is no longer infectios or ssceptible to frther bots of the disease; e describe it as remoed (R), since it is no an inert node in the contact netork that can no longer either catch or transmit the disease. This describes the fll model; e refer to it as the SIR model, after the three disease states that nodes experience. Figre 21.2 shos an example of the SIR model nfolding on a particlar contact netork throgh sccessie steps; in each step, shaded nodes ith dark borders are in the I state and shaded nodes ith thin borders are in the R state. The SIR model is clearly most appropriate for a disease that each indiidal only catches once in their lifetime; after being infected, a node is remoed either becase it has acqired lifetime immnity or becase the disease has killed it. In the next section, e ill consider a related model for diseases that can be caght mltiple times by the same person. Notice also that the branching process model from Section 21.2 is a special case of the SIR model: it simply corresponds to the SIR model here t I = 1 and the contact netork is an infinite tree, ith each node connected to a fixed nmber of neighbors in the leel belo.

8 652 CHAPTER 21. EPIDEMICS t t s y s y x z x z r r (a) (b) t t s y s y x z x z r r (c) (d) Figre 21.2: The corse of an SIR epidemic in hich each node remains infectios for a nmber of steps eqal to t I = 1. Starting ith nodes y and z initially infected, the epidemic spreads to some bt not all of the remaining nodes. In each step, shaded nodes ith dark borders are in the Infectios (I) state and shaded nodes ith thin borders are in the Remoed (R) state. Extensions to the SIR model. Althogh the contact netork in the general SIR model can be arbitrarily complex, the disease dynamics are still being modeled in a simple ay. Contagion probabilities are set to a niform ale p, and contagiosness has a kind of on-off property: a node is eqally contagios for each of the t I steps hile it has the disease. Hoeer, it is not difficlt to extend the model to handle more complex assmptions. First, e can easily captre the idea that contagion is more likely beteen certain pairs of nodes by assigning a separate probability p, to each pair of nodes and for hich links to in the directed contact netork. Here, higher ales of p, correspond to closer contact and more likely contagion, hile loer ales indicate less intensie contact. We can also choose to model the infectios period as random in length, by assming that an infected node has a probability q of recoering in each step hile it is infected, hile leaing

9 21.3. THE SIR EPIDEMIC MODEL 653 Figre 21.3: In this netork, the epidemic is forced to pass throgh a narro channel of nodes. In sch a strctre, een a highly contagios disease ill tend to die ot relatiely qickly. the other details of the model as they are. More elaborate extensions to the model inole separating the I state into a seqence of seeral states (e.g. early, middle, and late periods of the infection), and alloing the contagion probabilities to ary across these states [238]. This cold be sed, for example, to model a disease ith a highly contagios incbation period, folloed by a less contagios period hile symptoms are being expressed. Researchers hae also considered ariations on the SIR model in hich the disease-casing pathogen is mtating (and ths changing its disease characteristics) oer the corse of the otbreak [183]. The Role of the Basic Reprodctie Nmber. We no discss some obserations abot the SIR model, focsing on the most basic ersion of the model in an arbitrary netork. First, let s recall the claim made at the end of Section 21.2, that in netorks that do not hae a tree strctre, the simple dichotomy in epidemic behaior determined by the basic reprodctie nmber R 0 does not necessarily hold. In fact, it is not hard to constrct an example shoing ho this dichotomy breaks don. To do this, let s start ith the netork depicted in Figre 21.3, and sppose that these layers of to nodes at a time contine indefinitely to the right. Let s consider an SIR epidemic in hich t I = 1, the infection probability p is 2/3, and the to nodes at the far left are the ones that are initially infected. When e don t hae a tree netork, e need to decide ho to define an analoge of the basic reprodctie nmber. In a netork as highly strctre as the one in Figre 21.3, e can ork directly from the definition of R 0 as the expected nmber of ne cases of the disease cased by a single indiidal. (For less strctred netorks, one can consider R 0 to be the expected nmber of ne cases cased by a randomly chosen indiidal from the poplation.) In Figre 21.3, each infected node has edges to to nodes in the next layer; since it infects each ith probability 2/3, the expected nmber of ne cases cased by this

10 654 CHAPTER 21. EPIDEMICS t s y x z r Figre 21.4: An eqialent ay to ie an SIR epidemic is in terms of percolation, here e decide in adance hich edges ill transmit infection (shold the opportnity arise) and hich ill not. node is 4/3. So in or example, R 0 > 1. Despite this, hoeer, it is easy to see that the disease ill die ot almost srely after reaching only a finite nmber of steps. In each layer, there are for edges leading to the next layer, and each ill independently fail to transmit the disease ith probability 1/3. Therefore, ith probability (1/3) 4 =1/81, all for edges ill fail to transmit the disease and at this point, these for edges become a roadblock garanteeing the disease can neer reach the portion of the netork beyond them. Ths, as the disease moes along layer-by-layer, there is a probability of at least 1/81 that each layer ill be its last. Therefore, ith probability 1, it mst come to an end after a finite nmber of layers. This is a ery simple example, bt it already indicates ho different netork strctres can be more or less condcie to the spread of a disease een taking contagiosness and other disease properties as gien. Whereas the contact netork of the simple branching process from Section 21.2 as a tree that expanded rapidly in all directions, the netork in Figre 21.3 forces the disease to pass throgh a narro channel in hich a small breakdon in contagion can ipe it ot. Understanding ho specific types of netork strctre interact ith disease dynamics remains a challenging research qestion, and one that affects predictions abot the corse of real epidemics.

11 21.3. THE SIR EPIDEMIC MODEL 655 SIR Epidemics and Percolation. Ths far e hae been thinking abot SIR epidemics as dynamic processes, in hich the state of the netork eoles step-by-step oer time. This captres the temporal dynamics of the disease itself as it spreads throgh a poplation. Interestingly, hoeer, there is an eqialent and completely static ie of these epidemics that is often ery sefl from a modeling point of ie [44, 173]. We no describe ho to arrie at this static ie of the process, focsing on the basic SIR model in hich t I = 1. Consider a point in an SIR epidemic hen a node has jst become infectios, and it has a ssceptible neighbor. Node has one chance to infect (since t I = 1), and it scceeds ith probability p. We can ie the otcome of this random eent as being determined by flipping a coin that has a probability p of coming p heads, and obsering the otcome. From the point of ie of the process, it clearly does not matter hether the coin as flipped at the moment that first became infectios, or hether it as flipped at the ery beginning of the hole process and is only being reealed no. Contining this reasoning, e can in fact assme that for each edge in the contact netork from a node to a node a coin ith heads probability p is flipped at the ery beginning of the process (independently of the coins for all other pairs of neighbors), and the reslt is stored so that it can be later checked in the eent that becomes infectios hile is ssceptible. With all the coins flipped in adance, the SIR process can be ieed as follos. The edges in the contact netork for hich the coin flip is sccessfl are declared to be open; the remaining edges are declared to be blocked. The sitation is no as pictred in Figre 21.4, hich shos a sample reslt of coin flips consistent ith the pattern of infections in the example from Figre And e can no see ho to make se of the open and blocked edges to represent the corse of the epidemic: A node ill become infected dring the epidemic if and only if there is a path to from one of the initially infected nodes that consists entirely of open edges. Ths, hile Figre 21.4 looks sperficially different from the seqence of stages in Figre 21.2, it is in fact a beatiflly compact ay to smmarize the corse of the epidemic: the nodes that are eentally infected are precisely those that can be reached from the initially infected nodes along a seqence of open edges in the netork. This static ie of the model is often referred to as percolation, de to the folloing physical analogy. If e think of the contact netork as a system of pipes, and the pathogen as a flid moing throgh these pipes, then the edges in the contact netork on hich contagion scceeds are the open pipes and the edges on hich it fails are the blocked pipes. We no ant to kno hich nodes the flid ill reach, gien that it can only pass throgh open pipes. In fact, this is not simply an illstratie metaphor; percolation is a topic that has been extensiely stdied by physicists and mathematicians as a model for the flo of flids throgh certain types of poros media [69, 173]. It is both an interesting topic in its on right, and sefl for its role as an eqialent ie of the progress of an epidemic.

12 656 CHAPTER 21. EPIDEMICS (a) (b) (c) (d) (e) Figre 21.5: In an SIS epidemic, nodes can be infected, recoer, and then be infected again. In each step, the nodes in the Infectios state are shaded The SIS Epidemic Model In the preios sections e hae been considering models for epidemics in hich each indiidal contracts the disease at most once. Hoeer, a simple ariation on these models allos s to reason abot epidemics here nodes can be reinfected mltiple times. To represent sch epidemics, e hae nodes that simply alternate beteen to possible states: Ssceptible (S) and Infectios (I). There is no Remoed state here; rather, after a node is done ith the Infectios state, it cycles back to the Ssceptible state and is ready to catch the disease again. Becase of this alternation beteen the S and I states, e refer to the model as the SIS model. Aside from the lack of an R state, the mechanics of the model follo the SIR process ery closely. Initially, some nodes are in the I state and all others are in the S state. Each node that enters the I state remains infectios for a fixed nmber of steps t I. Dring each of these t I steps, has a probability p of passing the disease to each of its ssceptible neighbors. After t I steps, node is no longer infectios, and it retrns to the S state. Figre 21.5 shos an example of the SIS model nfolding on a three-node contact netork ith t I = 1. Notice ho node starts ot infected, recoers, and later becomes infected again e can imagine this as the contact netork ithin a three-person apartment, or a three-person family, here people pass a disease on to others they re liing ith, and then get it back from them later. As ith the SIR model, the SIS model can be extended to handle more general kinds of assmptions: different contagion probabilities beteen different pairs of people; probabilistic

13 21.4. THE SIS EPIDEMIC MODEL 657 recoery from the disease, in hich each infected node transitions back to the ssceptible state ith probability q each step; and mltiple stages of infection, ith arying disease properties across them. Life Cycles of SIR and SIS Epidemics. The examples in this section and preceding one sggest that the oerall trajectories of SIR and SIS epidemics on (finite-size) graphs are qalitatiely qite different. An SIR epidemic on a finite graph is brning throgh a bonded spply of nodes since nodes can neer be reinfected and therefore it mst come to an end after a relatiely small nmber of steps. An SIS epidemic, on the other hand, can rn for an extremely long time as it cycles throgh the nodes potentially mltiple times. Bt as Figre 21.5(e) illstrates, if there eer comes a point in an SIS epidemic hen all nodes are simltaneosly free of the disease, then the epidemic has died foreer: there are no longer any infected indiidals to pass the disease to others. And on a finite graph, there ill eentally (ith probability 1) come a point in time hen all contagion attempts simltaneosly fail for t I steps in a ro, and at this point it ill be oer. Ths a key qestion ith an SIS epidemic on a gien contact netork is to nderstand ho long the otbreak ill last, and ho many indiidals ill be affected at different points in time. For contact netorks here the strctre is mathematically tractable, researchers hae in fact proed knife-edge reslts for the SIS model similar to or dichotomy for branching processes. These reslts, on particlar classes of contact netorks, sho that at a particlar critical ale of the contagion probability p, an SIS epidemic on the netork ill ndergo a rapid shift from one that dies ot qickly to one that persists for a ery long time [52, 278]. This type of analysis tends to be mathematically qite complex, ith this critical ale of the contagion probability p depending in sbtle ays on the strctre of the netork. A Connection Beteen SIR and SIS Epidemics. Despite the differences beteen the SIR and SIS models, in fact it is possible to represent some of the basic ariants of the SIS model as special cases of the SIR model. This srprising relationship is frther eidence of the flexibility of the basic epidemic models, in hich formalisms defined in different ays trn ot to hae ery close connections to each other. We describe the relationship for the SIS model ith t I = 1, hen each node is infectios for a single step before recoering. The key insight is that if e think abot a node as in fact being a different indiidal at each time step, then e can represent things so that nodes are neer reinfected. Specifically, gien an instance of the SIS model ith t I = 1, e create a separate copy of each node for each time step t =0, 1, 2, 3 and onard. We ill call this the time-expanded contact netork. No, for each edge in the original contact netork, linking a node to a node, e create edges in the time-expanded contact netork from the copy of at time t to the copy of at time t + 1; this simply encodes the idea that

14 658 CHAPTER 21. EPIDEMICS step 0 step 1 step 2 step 3 step 4 (a) To represent the SIS epidemic sing the SIR model, e se a time-expanded contact netork step 0 step 1 step 2 step 3 step 4 (b) The SIS epidemic can then be represented as an SIR epidemic on this time-expanded netork. Figre 21.6: An SIS epidemic can be represented in the SIR model by creating a separate copy of the contact netork for each time step: a node at time t can infect its contact neighbors at time t + 1. can potentially catch the disease at time t + 1 if is infected at time t. Figre 21.6(a) shos this constrction applied to the contact netork from Figre The point is that the same SIS disease dynamics that preiosly circlated arond in the original contact netork can no flo forard in time throgh the time-expanded contact netork, ith copies of nodes that are in the I state at time t prodcing ne infections in copies of nodes at time t + 1. Bt on this time-expanded graph e hae an SIR process, since any copy of a node can be treated as remoed (R) once its one time step of infection is oer; and ith this ie of the process, e hae the same distribtion of otcomes as the original SIS process. Figre 21.6(b) shos the corse of the SIR epidemic that corresponds to the SIS epidemic in Figre 21.5.

15 21.5. SYNCHRONIZATION Synchronization The models e e deeloped gie s a frameork for thinking abot arios broader isses in the spread of disease. We already encontered one of these isses in the dichotomy for branching processes, hich proided a formal basis for the sensitiity of otbreaks to small ariations in contagiosness, and for the crcial role of the basic reprodctie nmber. We no look at a related isse in the global dynamics of a disease the tendency of epidemics for certain diseases to synchronize across a poplation, sometimes prodcing strong oscillations in the nmber of affected indiidals oer time. Sch effects are ell-knon for diseases inclding measles [196, 213] and syphilis [195]. When looking at pblic-health data, it is natral to look at periodic oscillations in the nmber of cases of a disease and to try positing external cases for the effect. For example, cycles in the prealence of syphilis across the U.S. oer the past 50 years hae traditionally been attribted to large-scale societal changes, inclding changes in sexal mores and other forces [195]. While sch factors clearly play a role, recent research has shon that oscillations and synchronization oer time can in fact reslt largely from the contagion dynamics of the disease itself, and that similar patterns can be created in direct simlations of the disease sing the types of models e hae been considering here [195, 267]. We no describe ho sch effects can be prodced sing simple epidemic models. The crcial ingredients appear to be a combination of temporary immnity and long-range links in the contact netork. Roghly, long-range links prodce coordination in the timing of flare-ps across dispersed parts of the netork; hen these sbside, the temporary immnity prodces a netork-ide deficit in the nmber and connectiity of ssceptible indiidals, yielding a large trogh in the size of the otbreak that directly follos the peak from the earlier flare-ps. We no describe ho to make this intitie pictre concrete sing simple models. The SIRS Epidemic Model. The first step in prodcing a model ith oscillations is to allo the disease to confer temporary bt not permanent immnity on infected indiidals a featre of many real diseases. To do this, e combine elements of the SIR and SIS models in a simple ay, so that after an infected node recoers, it passes briefly throgh the R state on its ay back to the S state. We call the reslting model the SIRS model [267], since nodes pass throgh the seqence S-I-R-S as the epidemic proceeds. In detail, the model orks as follos. Initially, some nodes are in the I state and all others are in the S state. Each node that enters the I state remains infectios for a fixed nmber of steps t I. Dring each of these t I steps, has a probability p of passing the disease to each of its ssceptible neighbors.

16 660 CHAPTER 21. EPIDEMICS (The ne featre of the model.) After t I steps, node is no longer infectios. It then enters the R state for a fixed nmber of steps t R. Dring this time, it cannot be infected ith the disease, nor does it transmit the disease to other nodes. After t R steps in the R state, node retrns to the S state. For an SIRS epidemic, the corse of the disease throgh a poplation is clearly affected not jst by the qantities p and t I, bt also by the length t R of the temporary immnity that is conferred. Small-World Contact Netorks. Temporary immnity can prodce oscillations in ery localized parts of the netork, ith patches of immnity folloing large nmbers of infections in a concentrated area. Bt for this to prodce large flctations that can be seen at the leel of the fll netork, the flare-ps of the disease hae to be coordinated so that they happen at roghly the same time in many different places. A natral mechanism to prodce this kind of coordination is to hae a netork that is rich in long-range connections, linking otherise far-apart sections of the netork. This kind of strctre is familiar from or discssion of small-orld properties in Chapter 20. There, e considered netork models here many of the links ere local and clstered connecting nodes ith ery similar social and geographic characteristics, according to the principle of homophily hile some ere long-range links, corresponding to eak ties that link ery different parts of the netork. In Chapter 20 e focsed on the effect this kind of strctre has on the distances beteen nodes. Bt there is a closely related conseqence: long-range links make it possible for things that happen in one part of the netork to qickly affect hat is happening elsehere. Watts and Strogatz obsered the releance of small-orld properties to synchronization in their original paper on the topic [411], and Kperman and Abramson shoed ho it cold natrally lead to synchronization and oscillation in epidemics [267]. For their analysis they constrcted random netorks ith small-orld properties, in a manner ery similar to the grid-pls-random-edges constrction discssed in Chapter 20; instead of the ersion from that chapter, they more closely folloed the original constrction of Watts and Strogatz, in hich a ring netork is reired to prodce random shortcts [411]. Specifically, they started ith a graph in hich the nodes are arranged in a ring, and each node is connected to its neighbors for some nmber of steps in each direction. These are all homophilos links, in that they connect nodes that are ery close together on the ring. Then, independently ith some probability c, they trned each edge into a eak tie by reiring one end of it to a node chosen niformly at random. Ths, the probability c controls the fraction of links in the netork that sere as long-range eak ties. When the SIRS model is rn on this kind of netork, one finds ery different behaior depending on the ale of c, as indicated in Figre When c is ery small, disease

17 21.5. SYNCHRONIZATION 661 Figre 21.7: These plots depict the nmber of infected people oer time (the qantity n inf (t) on the y-axis) by SIRS epidemics in netorks ith different proportions of long-range links. With c representing the fraction of long-range links, e see an abscence of oscillations for small c (c =0.01), ide oscillations for large c (c =0.9), and a transitional region (c =0.2) here oscillations intermittently appear and then disappear. (Reslts and image from [267].) transmission throgh the netork occrs mainly ia the short-range local edges, and so flare-ps of the disease in one part of the netork neer become coordinated ith flare-ps in other parts. As c increases, these flare-ps start to synchronize, and since each brst prodces a large nmber of nodes ith temporary immnity, there is a sbseqent trogh as the disease has difficlty making its ay throgh the sparser set of aailable targets. For ery large ales of c (sch as c =0.9 in Figre 21.7), there are clear aes in the nmber of affected indiidals; for intermediate ales of c (sch as c = 0.2) one obseres interesting effects in hich the system achiees netork-ide synchronization for a period, and then seems to fall back ot of sync for reasons that are hard to qantify. These reslts sho ho fairly complex epidemic dynamics can arise from simple models of contagion and contact strctre. There are, hoeer, a nmber of interesting open qestions;

18 662 CHAPTER 21. EPIDEMICS the reslts discssed here hae been primarily fond throgh simlation, and analyzing the onset of synchronization mathematically in this model remains largely nexplored. Sychronization in Epidemic Data. It is possible to stdy these effects empirically and ealate proposed models sing extensie records of disease prealence that reach back many years. Grassly, Fraser, and Garnett [195] performed an instrctie comparison of syphilis and gonorrhea that illstrates a nmber of synchronization principles. The prealence of syphilis exhibits prominent oscillations on an 8-11-year cycle, hile gonorrhea exhibits ery little in the ay of periodic behaior. Yet the to diseases affect similar poplations, and are presmably sbject to ery similar societal forces. These differences are consistent, hoeer, ith the fact that syphilis confers limited temporary immnity after infection, hile gonorrhea does not. Moreoer, the timing of the syphilis cycles fit ell ith the timing of the immne properties associated ith it. And from the cyclic patterns, one finds that the extent of synchronization beteen different regions of the United States increases oer time, sggesting that the contact netork on hich it spread became increasingly connected ith cross-contry links oer the second half of the 20th centry [195]. There are many frther directions in hich research on epidemic synchronization is proceeding, inclding attempts to model more complex temporal phenomena. For example, data for some diseases sch as measles shos that epidemics in different cities can synchronize so as to be ot of phase, ith the flare-ps in one city consistently coinciding ith troghs in the other [196]. One needs more than simply long-range contacts to explain sch properties [213]. There is also the qestion of ho immnization, preention programs, and other medical interentions can take adantage of these timing properties another ay in hich insights from een simple models can help to inform decision-making in this area Transient Contacts and the Dangers of Concrrency Ths far, or epidemic models hae taken the nderlying contact netork to be a relatiely static object, in hich all the links in the contact netork are present throghot the corse of the epidemic. This is a reasonable simplifying assmption for diseases that are relatiely contagios and spread qickly, at a rate faster than the typical creation or dissoltion of a contact. Bt as e moe don the spectrm toard diseases that spread throgh a poplation oer longer time scales, it is sefl to reisit these assmptions. For a disease like HIV/AIDS, the epidemic progresses oer many years, and its corse is heaily dependent on the properties of the sexal contact netork. Most people hae zero, one, or ery fe contacts at any single

19 21.6. TRANSIENT CONTACTS AND THE DANGERS OF CONCURRENCY 663 x x [1,5] [2,6] [1,5] [2,6] [7,11] [12,16] [12,16] [7,11] y y (a) In a contact netork, e can annotate the edges ith time indos dring hich they existed. (b) The same netork as in (a), except that the timing of the - and -y partnerships hae been reersed. Figre 21.8: Different timings for the edges in a contact netork can affect the potential for a disease to spread among indiidals. For example, in (a) the disease can potentially pass all the ay from to y, hile in (b) it cannot. point in time (a fe people hae many, hich is important as ell); and the identities of these contacts can shift significantly hile the disease progresses, as ne sexal partnerships are formed and others break p. So for modeling the contact netork in sch diseases, it is important to take into accont the fact that contacts are transient they do not necessarily last throgh the hole corse of the epidemic, bt only for particlar indos of time. Ths, e ill consider contact netorks in hich each edge is annotated ith the period of time dring hich it existed that is, the time range oer hich it as possible for one endpoint of the edge to hae passed the disease directly to the other. Figre 21.8(a) shos an example of this, ith the nmbers inside sqare brackets indicating the time ranges hen each edge exists. Ths the - and -x partnerships happen first, and they oerlap in time; after this, has a partnership ith and then later ith y. Note also that for this section in keeping ith the motiation from HIV/AIDS and similar diseases e assme the edges to be ndirected rather than directed, to indicate that infection can pass in either direction beteen a pair of people in a partnership. (As in preios sections, e cold also accomplish this by haing directed edges pointing in both directions beteen each pair of connected people, bt since eerything here ill be symmetric, it is more conenient to se ndirected edges.) The Conseqences of Transient Contacts. A little experimentation ith the example in Figre 21.8(a) indicates ho the timing of different edges can affect the spread of a disease.

20 664 CHAPTER 21. EPIDEMICS [1,5] [6,10] [1,5] [2,6] (a) s to partnerships happen serially (b) s to partnership s happen concrrently Figre 21.9: A disease tends to be able to spread more idely ith concrrent partnerships (b) than ith serial partnerships (a). For example, if has the disease at time 1, it is possible for it to spread all the ay to y, throgh and as intermediaries. (Of corse, if contagion is probabilistic as before, it ill not necessarily scceed in spreading; bt it has the potential to do so.) On the other hand, cannot spread the disease to x: node cold pass the disease to, ho cold pass it to ; bt by the time it reaches, the partnership of and x is long oer. Moreoer, changing the timing of partnerships can change the possible transmission pathays, een as the set of nderlying contacts remains the same. For instance, the example in Figre 21.8(b) differs from the one in Figre 21.8(a) only in that the temporal order of the - and -y partnerships has been reersed. Bt notice that hile as able to pass the disease all the ay to y in Figre 21.8(a), it cannot do so in Figre 21.8(b): in the latter case, the -y partnership is oer by the time the disease cold possibly get from to. Sch considerations are crcial as health orkers and epidemiologists map ot the contact netorks associated ith a disease sch as HIV/AIDS. For example, e can see from the difference beteen Figres 21.8(a) and 21.8(b) that in order for y to kno hether he or she is at risk from a disease carried by, it is not enogh een to map ot the fll set of sexal partnerships; it is crcial to kno information abot the order of eents as ell. Or if e go back to the striking Figre 2.7 from Chapter 2, mapping ot the relationships ithin a high school, e can appreciate that the image itself is not enogh to flly chart the potential spread of diseases throgh this poplation e old also need to kno the timing of these relationships. Netorks in hich the edges only exist for specific periods of time hae been the sbject of modeling efforts in many areas, inclding sociology [182, 305, 258], epidemiology [307, 406], mathematics [106], and compter science [53, 239]. It is an isse that is releant not jst to the spread of disease, bt also to a ide range of settings that are modeled by netorks. For example, the diffsion of information, ideas, and behaiors throgh social netorks clearly also depends on ho the timing of different commnications beteen people either enables or blocks the flo of information to different parts of the poplation. Concrrency. Differences in the timing of contacts do not jst affect ho has the potential to spread a disease to hom; the pattern of timing can inflence the seerity of the oerall

21 21.6. TRANSIENT CONTACTS AND THE DANGERS OF CONCURRENCY 665 x x [1,5] [2,6] [1,5] [2,6] [12,16] [3,7] [7,11] [1,5] y y (a) No node is inoled in any concrrent partnerships (b) All partnerships oerlap in time Figre 21.10: In larger netorks, the effects of concrrency on disease spreading can become particlarly prononced. epidemic itself. A timing pattern of particlar interest and concern to HIV researchers is concrrency [307, 406]. A person is inoled in concrrent partnerships if he or she has to or more actie partnerships that oerlap in time. For example, in each of Figres 21.9(a) and 21.9(b), node has partnerships ith each of and. Bt in the first of these figres, the partnerships happen serially first one, then the other hile in the second, they happen concrrently, oerlapping in time. The concrrent pattern cases the disease to circlate more igorosly throgh this three-person netork. and may not be aare of each other s existence, bt the concrrent partnerships make it possible for either of or to spread the disease to the other; the serial partnerships only allo spreading from to, bt not the other ay. In larger examples one can find more extreme effects; for example, Figre 21.10(b) differs from Figre 21.10(a) only in that the time indos of the partnerships hae been pshed together so that they all oerlap. Bt the effect is considerable: here the pattern in Figre 21.10(a) alloed different parts of the netork to be alled off from each other by the timing effects, the concrrent partnerships make it possible for any node ith the disease to potentially spread it to any other. In simlations ith arios notions of concrrency, Morris and Kretzschmar fond that small changes in the amont of concrrency keeping other ariables like the aerage nmber and dration of partnerships fixed cold prodce large changes in the size of the epidemic [307]. Qalitatiely, this aligns ell ith the intition from earlier sections, that changing the aerage nmber of ne cases of a disease cased by an infected indiidal een slightly can sometimes hae significant conseqences. For some of the simplest models, sch as the branching process, it is possible to make this intition precise; for more complex

22 666 CHAPTER 21. EPIDEMICS models sch as the present one inoling concrrency in arbitrary netorks, it remains the topic of ongoing research. Concrrency is jst one particlar kind of pattern to be fond in the timing of relationships in a contact netork. Frther research in this area cold possibly ncoer more sbtle patterns as ell; the interaction of timing and netork strctre has the potential to proide frther insights into the ay diseases spread throgh the changing contacts ithin a poplation Genealogy, Genetic Inheritance, and Mitochondrial Ee Or discssion of epidemics has proided s ith a ay of thinking abot processes that spread randomly oer time throgh a netork. As mentioned earlier, this is a sefl frameork for modeling many kinds of things that spread, not jst diseases. The spread of information can be modeled this ay, as an alternatie to the approaches based on explicit decision rles discssed in Chapter 19. In sch settings, adapting the ideas from this chapter can be relatiely straightforard bt still ery informatie. In this section, instead, e apply the perspectie of random spreading to a sitation here the connection is at first a bit more sbtle; it takes a little ork to precisely identify the netork and the process that is spreading throgh it. The setting is that of genetic inheritance. What e ill find is that hen e ie inheritance of traits as a random process that takes place on a netork linking organisms in sccessie generations in other ords, ith edges connecting parents to their offspring then e can obtain insight into some fndamental hereditary processes. We start ith a story that illstrates some of the basic genetic isses e ll consider. Mitochondrial Ee. In 1987, Rebecca Cann, Mark Stoneking, and Allan Wilson pblished a paper in the jornal Natre [94] here they proided eidence for a rather striking proposition. Consider folloing yor maternal ancestry backard in time throgh hman history, prodcing a trail that goes from yo to yor mother, to her mother (i.e. yor maternal grandmother), to her mother, and so on indefinitely. Each of s in principle can prodce sch a maternal ancestry trail, hich e ll call a maternal lineage. No, the claim of Cann, Stoneking, and Wilson as that all these lineages in fact meet at a single oman ho lied beteen 100,000 and 200,000 years ago, probably in Africa. She is at the root of all or maternal ancestries. Let s first ask ho they reached this conclsion, and then consider hat it signifies. One ay to infer facts abot maternal ancestries is to stdy the DNA fond not in or cells nclei, bt in the mch smaller, separate genome that each of s has in or cells

23 21.7. GENEALOGY, GENETIC INHERITANCE, AND MITOCHONDRIAL EVE 667 mitochondria. Unlike nclear DNA, hich contains parts of both or parents genomes, this mitochondrial DNA is (to a first approximation) passed to children entirely from their mothers. So roghly speaking, aside from random mtations, yo hae yor mother s DNA, she has her mother s DNA, and so on throgh yor maternal ancestry. With this in mind, Cann, Stoneking, and Wilson analyzed the mitochondrial DNA of people dran from a ide sample of geographic and ethnic backgronds; sing standard techniqes to estimate the rate at hich genetic seqences ill dierge throgh random mtations oer many generations, they conclded that all the mitochondrial DNA in this poplation likely had a common origin roghly 100, ,000 years ago. By common origin here, e mean a single mitochondrial genome belonging to a single hman being; becase she is the sorce of the mitochondrial DNA of eeryone on earth, researchers standardly refer to this oman as Mitochondrial Ee. This finding caght the pblic imagination hen it as first annonced; it receied a fair amont of media attention at the time, and its implications hae been nicely explored in general books abot hman history [333]. The analysis inoled in the original finding has since been refined by a nmber of other research grops; caeats hae been introdced de to the fact that the inheritance of mitochondrial DNA may be more complicated than originally thoght; bt the basic conclsion has been mainly accepted at a general leel. As to hat this finding signifies: on first hearing, it takes a bit of thoght to sort ot hat it implies and hat it doesn t. It is indeed striking to be able to posit the existence of a single person from the not-so-distant eoltionary past ho is an ancestor of eeryone. Mitochondrial Ee (in contrast to her namesake Ee from the Bible) as not asserted to be the only liing oman in her time; there ere presmably many other omen liing at the same time as her, bt from the point of ie of present-day mitochondrial DNA, all these omen are genetically irreleant: somehere along the line from then to no, each of their lines of mitchondrial DNA died ot. On the other hand, one also needs to be carefl before attribting too mch to the relatiely recent existence of Mitochondrial Ee. In particlar, hile her contemporaries ere genetically irreleant to or mitochondrial DNA, they are not irreleant to the remainder of or genomes; each of s has genetic contribtions from a large nmber of ancestors. (Thogh een here there is more going on than meets the eye, as e ll discss shortly.) Moreoer, the oerlapping patterns of or respectie ancestries are complex and still not ell nderstood; hat e learn from Mitochondrial Ee is that all or ancestries are pinned together along their maternal lines, a cople of hndred thosand years into the past. Ultimately, the identification of Mitochondrial Ee as in a sense a shocase of ideas that had been emerging in the genetics commnity oer the preios decade [245, 325]. These ideas ere based on models that cold predict the existence of common ancestors and make estimates abot their recency. And they shoed that at a mathematical leel,

24 668 CHAPTER 21. EPIDEMICS crrent generation ne generation each offspring comes from a single parent chosen niformly at random Figre 21.11: In the basic Wright-Fisher model of single-parent ancestry, time moes stepby-step in generations; there are a fixed nmber of indiidals in each generation; and each offspring in a ne generation comes from a single parent in the crrent generation. independent of the difficlty of establishing eidence from genetic data, the existence of someone like Mitochondrial Ee as not only natral, bt in fact as e ill see next essentially ineitable. At their core, these models ere bilt from a probabilistic formalism inoling netorks; indeed, een in a qalitatie sense, one can appreciate something epidemic-like abot the ay in hich copies of different people s mitochondrial DNA spread throgh sbseqent generations, inhabiting ftre offspring, ntil one eentally crods ot all the others. We no describe the basic ersions of these models, and ho they connect to qestions abot ancestry. A Model of Single-Parent Ancestry We se a fndamental model of ancestry knon in poplation genetics as the Wright-Fisher model [325]. To remain tractable, the model inoles a nmber of simplifying assmptions. Consider a poplation that is constrained by resorces to maintain a fixed size N in each generation. Time moes step-by-step from one generation to the next; each ne generation is formed by haing the crrent set of N indiidals prodce N offspring in total. Each offspring in this ne generation is prodced from a single parent, and this parent is selected independently and niformly at random from among those in the crrent generation. Figre depicts this process; as shon there, e can dra the relationship of one generation to the next as a graph, ith a node for each indiidal, and an edge connecting each offspring to their parent chosen niformly at random from the preios generation. Notice that becase of this rle for selecting parents, certain indiidals in the pper generation can hae mltiple children (sch as the first and last in Figre 21.11), hile others may hae none.

25 21.7. GENEALOGY, GENETIC INHERITANCE, AND MITOCHONDRIAL EVE 669 s t x y z Figre 21.12: We can rn the model forard in time throgh a seqence of generations, ending ith a set of present-day indiidals. Each present-day indiidal can then follo its single-parent lineage by folloing edges leading pard throgh the netork. The strctre of this model reflects a fe nderlying assmptions. To begin ith, e re assming a netral model in hich no indiidal has a selectie adantage in reprodction; eeryone has the same chance of prodcing offspring. Frthermore, e re modeling a sitation in hich each indiidal is prodced from a single parent, as opposed to to parents in a sexally reprodcing poplation. This is consistent ith seeral possible interpretations. First, and most directly, it can be sed to model species that engage in asexal reprodction, ith each organism arising from a single parent. Second, it can be sed to model single-parent inheritance een in sexally reprodcing poplations, inclding the inheritance of mitochondrial DNA among omen as in or discssion aboe. In this interpretation, each node represents a hman oman, ith omen linked to their mothers in the preios generation. Moreoer, as e ill discss later, there is in fact a mch more general ay to se this model to think abot inheritance in sexally reprodcing poplations.

26 670 CHAPTER 21. EPIDEMICS s t x y z Figre 21.13: A re-draing of the single-parent netork fom Figre As e moe back in time, lineages of different present-day indiidals coalesce ntil they hae all conerged at the most recent common ancestor. Third, it can be sed to model prely social forms of inheritance, sch as masterapprentice relationships. For example, if yo receie a Ph.D. in an academic field, yo generally hae a single primary adisor. If yo model stdents as being descended from adisors, than e can trace ancestries throgh seqences of adisors back into the past jst as e traced maternal lineages. No, if e rn this model forard in time throgh mltiple generations, e get a netork sch as the one pictred in Figre Each indiidal is connected to one parent in the preios generation; time rns from top to bottom, ith N present-day indiidals in the loest layer (named s throgh z in the figre). Notice that from any one of these indiidals at the bottom, e can trace its single-parent lineage backard in time by folloing edges pard, alays taking the single edge leading p ot of each node e enconter. If e imagine the indiidals in the bottom ro of Figre to be present-day omen, then Mitochondrial Ee old be the loest node in the figre here all the maternal lineages first flly conerge. It s a bit tricky, isally, to find this node in Figre 21.12, bt e can re-

27 21.7. GENEALOGY, GENETIC INHERITANCE, AND MITOCHONDRIAL EVE 671 dra the same ancestries ith the paths nscrambled in Figre 21.13, and then the location of Ee becomes easy to see: she s the third node in the second ro from the top (ith the lineages leading back to her consisting of the darkened edges). These examples indicate ho the existence of common ancestors and een the nmber of generations needed to reach them can be predicted from the Wright-Fisher model. To do this, e employ a sefl trick for reasoning abot the model: e think of the ancestries as being bilt backard in time, rather than forard. In other ords, an eqialent ie of the model is to take a set of present-day indiidals, and constrct earlier generations one at a time by haing each crrent indiidal choose its parent in the preios generation independently and niformly at random. We can see ho this orks by moing pard throgh the leels of Figre Wheneer to indiidals happen to choose the same parent, then their lineages coalesce into a common lineage from that point onard. Ths, e start ith N distinct lineages in the present, bt as e bild generations going backard in time, the nmber of distinct lineages decreases heneer indiidals on distinct lineages choose the same parent. This coalescence ill happen rapidly at first, hen there are many lineages and the probability of a collision beteen lineages is high; as time goes on, the nmber of distinct lineages of present-day indiidals shrinks more and more sloly. Bt heneer there is more than one distinct lineage, there is a finite expected time ntil to of them collide, and so the process mst eentally reach a single lineage. The node at hich this first happens is called the most recent common ancestor the analoge of Mitochondrial Ee in this model. The model is simple enogh that one can estimate the expected time ntil the collisions among lineages, and hence the expected nmber of generations to the most recent common ancestor [245, 325]. Genetic Interpretations. Althogh the maternal inheritance of mitochondrial DNA makes for a ery simple single-parent process, the Wright-Fisher model is releant to sexally reprodcing poplations for a mch more fndamental reason. While the chromosomes of yor parents recombined to prodce yor genome, making yor chromosomes a patchork of theirs, any single point in yor genome a single ncleotide on one of yor chromosomes as inherited from jst one of yor mother or father. They, in trn, inherited it from jst one of their mother or father, and so on. As a reslt, if e ant to trace the ancestry of a single point in yor genome, e are folloing a single-parent lineage, een thogh offspring are prodced by sexal reprodction. The most recent common ancestor for this particlar point, looking across a poplation of N indiidals, ill ths follo from the same analysis e e seen aboe, as it did for mitochrondrial DNA. Becase of recombination, the lineages for one point in the genome may differ from the lineages for een a nearby point, and hence the most recent common ancestors may differ as ell. One can deelop probabilistic models for ho these lineages relate to each other, bt

28 672 CHAPTER 21. EPIDEMICS the analysis becomes mch more complex [418]. There are many other isses that arise hen extending these simplified models to more complex genetic applications. For example, geographic barriers in a poplation can isolate indiidals from each other, and this can hae an effect on the patterns of interaction among lineages [354]. More generally, spatial constraints on the interactions among indiidals can affect these patterns, proiding another setting in hich netork properties can potentially inform broader conclsions abot genetic otcomes Adanced Material: Analysis of Branching and Coalescent Processes In this section, e analyze to of the basic processes discssed in this chapter: the branching process for the spread of an epidemic ith simplified contact netork strctre, and the coalescent process for the merging of lineages back to a common ancestor. Both of these are based on probabilistic reasoning inoling branching tree strctres: the first as the epidemic spreads forard throgh indiidals, and the second as the lineages trael backard in time. A. Analysis of Branching Processes Recall the branching process model that e considered in Section 21.2: each infected indiidal meets k others and infects each ith probability p. Ths, the expected nmber of ne cases of the disease cased by each infected indiidal is R 0 = pk, the basic reprodctie nmber. We ant to sho that the persistence of the disease depends critically on hether R 0 is smaller or larger than 1, a notion that e ill formlate as follos. Recall that the poplation in this model is organized into a tree (as shon in Figre 21.1(a)) in hich eery node is connected to k nodes jst belo it. Let q n denote the probability that the epidemic sries for at least n aes in other ords, that some indiidal in the n th leel of the tree becomes infected. Let q be the limit of q n as n goes to infinity; e can think of this as the probability that the disease persists indefinitely. We ill proe the folloing claim. Claim: (a) If R 0 < 1 then q =0. (b) If R 0 > 1 then q > 0. This establishes the knife-edge qality of R 0 that e discssed in Section The Expected Nmber of Infected Indiidals. We start by considering an approach to this problem that gets s partay to a proof of the claim: considering the expected nmber of infected indiidals at each leel of the tree. First, let s consider the total nmber of indiidals at each leel. The nmber of indiidals at any gien leel exceeds the nmber at the preios leel by a factor of k, and

29 21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES673 j Indiidal j is infected if each contact from the root to j sccessflly transmits the disease Figre 21.14: To determine the probability that a particlar node is infected, e mltiply the (independent) probabilities of infection on each edge leading from the root to the node. therefore the nmber ho are at leel n is k n. (This is also tre at leel n = 0: the top leel consists of jst the root, and k 0 = 1.) No, let X n be a random ariable eqal to the nmber of infected indiidals at leel n. One ay to think abot the expected ale E [X n ] is to rite X n as a sm of simpler random ariables as follos. For each indiidal j at leel n, let Y nj be a random ariable eqal to 1 if j is infected, and eqal to 0 otherise. Then X n = Y n1 + Y n2 + + Y nm, here m = k n, since the right-hand side simply conts p, one-by-one, the nmber of infected indiidals at leel n. Linearity of expectation says that the expectation of the sm of a set of random ariables is eqal to the sm of their expectations, and so E [X n ]=E [Y n1 + Y n2 + + Y nm ]=E [Y n1 ]+E [Y n2 ]+ + E [Y nm ]. (21.1) The reason to rite things this ay is that each expectation on the right-hand side is extremely easy to ork ot: E [Y nj ]=1 Pr [Y nj = 1] + 0 Pr [Y nj = 0] = Pr [Y nj = 1], and so the expectation of each Y nj is jst the probability that indiidal j gets infected.

30 674 CHAPTER 21. EPIDEMICS probability p n of each being infected n k indiidals Figre 21.15: The expected nmber of indiidals infected at leel n is the prodct of the nmber of indiidals at that leel (k n ) and the probability that each is infected (p n ). Indiidal j at depth n gets infected precisely hen each of the n contacts leading from the root to j sccessflly transmit the disease, as shon in Figre Since each contact transmits the disease independently ith probability p, indiidal j is infected ith probability p n. Therefore E [Y nj ] = p n. We hae already conclded that there are k n indiidals at leel n of the tree, and hence k n terms on the right-hand side of Eqation (21.1). Therefore, as smmed p in Figre 21.15, e conclde that E [X n ]=p n k n =(pk) n = R n 0. (21.2) From Expected Vales to Probabilities of Persistence. Eqation (21.2) sggests the importance of the basic reprodctie nmber R 0 in reasoning abot the spread of an epidemic in the branching process model. No let s consider hat this tells s abot q, the probability that the epidemic persists indefinitely. First, the fact that E [X n ]=R0 n immediately establishes part (a) of the Claim, that hen R 0 < 1 e hae q = 0. To see hy, e go back to the definition of E [X n ] and apply a fact that e also fond sefl in Section To recap the discssion there, the definition for the expected ale is E [X n ]=1 Pr [X n = 1] + 2 Pr [X n = 2] + 3 Pr [X n = 3] + (21.3)

31 21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES675 and an alternate bt eqialent ay to rite the right-hand side is as Pr [X n 1] + Pr [X n 2] + Pr [X n 3] + (21.4) since e obsere that each term Pr [X n = i] contribtes exactly i copies of itself to the sm in (21.4). Therefore e hae E [X n ] = Pr [X n 1] + Pr [X n 2] + Pr [X n 3] + (21.5) From (21.5) e obsere that E [X n ] mst be at least as large as the first term on the right-hand side, and so E [X n ] Pr [X n 1]. Notice also that Pr [X n 1] is precisely the definition of q n, and so E [X n ] q n. Bt E [X n ]=R n 0 hich is conerging to 0 as n gros, and hence q n mst also be conerging to 0. This shos that q = 0 hen R 0 < 1. No, hen R 0 > 1, the expected ales E [X n ]=R n 0 go to infinity as n gros. Hoeer, this fact by itself is not enogh to sho that q > 0. It is entirely possible to hae a seqence of random ariables for hich E [X n ] goes to infinity bt Pr [X n > 0] conerges to 0 as n gros. (As a simple example, sppose that X n ere a random ariable taking the ale 4 n ith probability 2 n, and taking the ale 0 otherise. Then E [X n ] = (4/2) n =2 n, hich goes to infinity, hile Pr [X n > 0] = 2 n, hich goes to 0.) This on t happen in or case, bt these considerations do say that to establish q > 0 hen R 0 > 1, e ll need to se something more specific abot the process than simply the expected nmber of infected indiidals. We do this no, deeloping a formla for q n that in the end ill allo s to determine the ale of q exactly. A Formla for q n. The qantity q n depends on three more fndamental qantities: the nmber of contacts per indiidal k, the contagion probability p, and the leel of the tree n. In fact, it s difficlt to rite don a direct formla for q n in terms of these qantities, bt it s not hard to express q n in terms of q n 1. This is hat e ll do first. Consider the root node, and let s first ask hat it old take for the folloing eent to hold: ( ) The disease spreads throgh the root node s first contact j and then contines to persist don to n leels in the part of the tree reachable throgh j. This is illstrated in Figre First, for the eent ( ) to hold, it old reqire that j catches the disease directly from the root, hich happens ith probability p. At this point, j becomes completely analogos to the root node of its on branching process, consisting of all nodes reachable from it donard in the tree. So for eent ( ) to hold, after j is infected, it is then necessary that the disease persists for n 1 leels in the ersion of the branching process in hich e ie node j as the root. This happens ith probability q n 1,

32 676 CHAPTER 21. EPIDEMICS prob p j prob. q n-1 n-1 leels Figre 21.16: In order for there to be an infection at leel n, the root mst infect one of its immediate descendants, and then this descendant mst, recrsiely, prodce an infection at leel n 1. by the definition of q n 1. Therefore, the probability of the eent ( ) is pq n 1. Or, taking the complementary ie, eent ( ) fails to hold ith probability 1 pq n 1. No, there is a copy of eent ( ) for each of the direct contacts of the root node, and each fails to hold ith probability 1 pq n 1. Since they re independent, the probability that they all fail to hold is (1 pq n 1 ) k. At this point, e re almost done. The disease fails to persist don to leel n of the tree, starting at the root, if it fails to reach leel n throgh any of the root s direct contacts. In other ords, the disease fails to persist to leel n precisely hen all the copies of eent ( ), for each direct contact of the root, fail to hold. Again, e jst determined that this probability is (1 pq n 1 ) k. Bt this probability is also 1 q n, since by the definition of q n, the qantity 1 q n is exactly the probability that the disease fails to persist to n leels. Therefore, 1 q n = (1 pq n 1 ) k and soling for q n e get q n =1 (1 pq n 1 ) k. (21.6)

33 21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES677 1 y = x y = f(x) 0 1 Figre 21.17: To determine the limiting probability of an infection at depth n, as n goes to infinity, e need to repeatedly apply the fnction f(x) =1 (1 px) k, hich is the basis for the recrrence q n = f(q n 1 ). Since e are assming that the root is infected, and e can treat the root as leel 0 of the tree, e hae q 0 = 1; this simply says that the root is infected ith probability 1. Starting from q 0 = 1, e can then bild p the ales q 1,q 2,q 3,... in order, determining each from the preios one in the list sing Eqation (21.6). Simply being able to determine the ales of each q n this ay, hoeer, doesn t immediately tell s here they re headed as n goes to infinity. For that e need a techniqe that looks at the limiting ale of this seqence. Folloing the ales q n to a limit. If e define the fnction f(x) =1 (1 px) k, then e can rite Eqation (21.6) as follos: q n = f(q n 1 ). This sggests a ery clean, prely algebraic ay of formlating or qestion abot q. We hae a fnction f(x) = 1 (1 px) k, and e simply ant to stdy the seqence of ales 1,f(1),f(f(1)),f(f(f(1))),..., obtained by applying f repeatedly. To get started thinking abot this, let s plot the fnction f on a pair of x-y axes, as in Figre Here are some basic facts abot f that help in prodcing this plot. First, f(0) = 0 and f(1) = 1 (1 p) k < 1. This means that the plot of f passes throgh the origin, bt lies belo the line y = x once x = 1, as shon in Figre Second, the deriatie of f is f (x) =pk(1 px) k. Notice that as x ranges beteen 0 and 1, the qantity f (x) is positie bt monotonically decreasing. This means that f has the increasing bt concae shape depicted in Figre

34 678 CHAPTER 21. EPIDEMICS 1 y = x y = f(x) 0 1 Figre 21.18: When e repeatedly apply the fnction f(x), starting at x = 1, e can follo its trajectory by tracing ot the seqence of steps beteen the cres y = f(x) and y = x. Finally, the slope of f at x = 0 is eqal to f (0) = pk = R 0. So in the case hen R 0 > 1, hich is hat e re focsing on no, the fnction f starts ot aboe the line y = x for small positie ales of x. When R 0 > 1, e can take these points together that y = f(x) starts ot aboe y = x for small positie ales of x bt ends p belo it by the time e get to x = 1 and conclde that y = f(x) mst cross y = x somehere in the interal beteen 0 and 1, at a point x > 0. No, sing this plot, let s take a geometric ie of the seqence of ales 1,f(1),f(f(1)),f(f(f(1))),... that e re analyzing. In particlar, let s track this seqence on the line y = x. If e re crrently at a particlar point (x, x) on the line y = x, and e ant to get to the point (f(x),f(x)), e can do that as follos. We first moe ertically to the cre y = f(x); this pts s at the point (x, f(x)). We then moe horizontally back to the line y = x; this pts s at the point (f(x),f(x)) as desired. This to-step ertical-horizontal motion is depicted as the first to parts of the dashed line in Figre Contining this process, e pass throgh all the points in the seqence x, f(x),f(f(x)),... along the line y = x. If e start this from x = 1, as indicated in Figre 21.18, the process conerges to the point (x,x ) here the line y = x meets the cre y = f(x). No e can go back to the interpretation of all this in terms of the branching process. The seqence of ales

35 21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES679 1 y = x y = f(x) 0 1 Figre 21.19: When y = f(x) only intersects y = x at zero, the repeated application of f(x) starting at x = 1 conerges to 0. 1,f(1),f(f(1)),... is precisely the seqence q 0,q 1,q 2,..., as e arged aboe, and so e hae conclded that it conerges to x > 0: the niqe point at hich f(x) =x in the interal strictly beteen 0 and 1. This concldes the argment that hen R 0 > 1, the probability that the epidemic persists for n leels conerges to a positie ale as n goes to infinity. It is also orth noticing that this style of analysis shos that q = 0 hen R 0 < 1. Indeed, hen R 0 < 1, the cre y = f(x) looks mch like it does in Figre 21.17, except that its deriatie at 0 is R 0 < 1, and so it lies ot belo the line y = x for the hole interal beteen 0 and 1. This means that hen e follo the seqence of ales 1,f(1),f(f(1)),... as the dashed lines do in Figre 21.18, it descends all the ay to x = 0 ithot stopping at any intermediate point. (See Figre ) This shos that the reslting limit, hich is q, is eqal to 0 in this case. B. Analysis of Coalescent Processes We no analyze a different process arising from earlier in the chapter the merging of ancestral lineages discssed in Section In particlar, e ill derie an estimate of the expected nmber of generations one has to go back in order to find the most recent common ancestor for a set of indiidals in the model from that section [245, 325]. Like the analysis of branching processes, this ill reqire probabilistic calclations on trees. In this case, hoeer, it is tricky to get an exact anser, and e ill make se of to approximations as

36 680 CHAPTER 21. EPIDEMICS Figre 21.20: We can ie the search for coalescence as a backard alk throgh a seqence of earlier generations, folloing lineages as they collide ith each other. e estimate the reqired nmber of generations. (In fact, these approximations still allo for a ery accrate estimate.) In addition to the approximations, hich e ll specify in context later, e start by arying the statement of the problem slightly, folloing the original ork on the topic. Specifically, e ill focs on a small sample of k indiidals in a large poplation of size N; rather than analyzing the time ntil all lineages in the fll poplation merge into a common ancestor, e ill consider the time ntil the lineages of these k merge into a common ancestor. This is reasonable from the point of ie of applications, since generally one is only eer stdying a fixed-size sample of a large poplation; also, the calclations inoled proide insight into the qestion for the fll poplation as ell. To recall the model from Section 21.7, adapted to the plan of looking at fixed-size samples

37 21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES681 of k indiidals, e can pose the qestion as follos. There are N indiidals in each generation. For each of the k indiidals in the initial sample, e choose a parent for each niformly at random from the preios generation. We contine orking backard in time this ay, extending each of the k lineages throgh earlier generations. Wheneer e get to a generation here to indiidals happen to choose the same parent, their lineages merge (since their ancestors ill no be the same), and so the process contines ith feer distinct lineages to track. Finally, e stop hen e first reach a point here the nmber of lineages has been redced to one, a moment that e call coalescence. We ant to estimate the expected time ntil coalescence occrs. Figre illstrates this on an example ith k = 6 initial present-day indiidals (in the bottom ro) and a poplation size of N = 27 (the nmber of nodes in each ro). The Probability that Lineages Collide in One Step. The key to this analysis is to consider a single step, in hich e hae a set of j distinct lineages that e re tracking, and e ant to estimate the probability that at least to of them ill choose a common parent in the preios generation. (We ill call this a collision beteen to lineages.) The easiest case to think abot is j = 2. Sppose e ie the random choice as being made seqentially by the to lineages nder consideration. The first lineage chooses a parent niformly at random, and then there is only a collision if the second lineage picks the same parent niformly at random from the N aailable choices. The probability that this occrs is therefore exactly 1/N. Things get more complicated hen j is larger than 2. First, let s compte the probability that no to lineages collide by imagining that the lineages choose their parents one at a time. For no to lineages to collide, it mst be the case that after the first lineage chooses a parent, the second chooses a parent distinct from this, the third chooses a parent distinct from these to, and so forth, p to the j th lineage, ho mst choose a parent distinct from the first j 1. The probability that this happens is ( 1 1 )( 1 2 )( 1 3 ) ( 1 j 1 ). N N N N Expanding ot this prodct, e see that it is eqal to ( ) j (terms ith N 2 or higher in the denominator). N In particlar, it is at most ( ) j 1 1 N + g(j) N 2 for a fnction g( ) that depends only on j. So far this calclation has been exact, bt here e come to the first of to approximations, folloing [245]: rather than deal ith the

38 682 CHAPTER 21. EPIDEMICS complexity of this last term, e obsere that hen the poplation size N is mch larger than j, expressions of the form g(j)/n 2 are negligible compared to ( j 1)/N. We therefore ignore them, and approximate the probability that no to lineages collide by ( ) j 1 1 =1 N j(j 1) 2N. (21.7) No, hen to lineages do in fact collide, there are a nmber of possibilities: it cold be that there is simply a to-ay collision beteen to of the lineages hile all the others remain distinct, or it cold be that more than to lineages collide in a single generation. We no describe ho the latter scenario can come to pass, and then arge that it is ery nlikely. First, it cold be that three lineages all choose the same parent in a single generation. For any particlar set of three lineages, the probability this happens is exactly 1/N 2 : imagining the choice being made seqentially, the first lineage can pick any parent, and then the second and third mst independently pick this same parent from the N aailable choices. No, since there are feer than j 3 sets of three lineages, the probability that any three-ay collision happens in a gien generation is less than j 3 /N 2. When N is mch larger than j, this qantity is negligible compared to expressions as in (21.7) that only hae N in the denominator. Alternately, it cold be that to different pairs of lineages each hae a separate, toay collision in the same generation: sppose that lineage A collides ith lineage B, and lineage C collides ith lineage D. The collision of A and B has probability 1/N, and the collision of C and D is an eent independent from this, also ith probability 1/N. Therefore, for this particlar choice of for lineages, both collisions happen ith probability 1/N 2. Since there are less than j 4 ays of choosing A, B, C, and D, the probability that there are simltaneos to-ay collisions inoling any choice of for lineages is less than j 4 /N 2. Again, ith N mch larger than j, this qantity is negligible compared to expressions that only hae N in the denominator. These argments lead to or second approximation: e ill assme there is neer a generation prior to the most recent common ancestor in hich e hae more than a single to-ay collision among lineages. This means that hen e hae a generation in hich the j crrent lineages fail to remain completely distinct, it happens becase exactly to of them choose a common parent, redcing the nmber of lineages from j to j 1. The Expected Time Until Coalescence. Or to approximations hae led to a ery clean ie of the process as it orks backard in time. In this approximate ie, e start

39 21.8. ADVANCED MATERIAL: ANALYSIS OF BRANCHING AND COALESCENT PROCESSES683 From 2 to 1: aiting for an eent of prob. 1/N From 3 to 2: aiting for an eent of prob. 3/N From 4 to 3: aiting for an eent of prob. 6/N From 5 to 4: aiting for an eent of prob. 10/N From 6 to 5: aiting for an eent of prob. 15/N Figre 21.21: Assming that no three lineages eer collide simltaneosly, the time to coalescence can be compted as the time for a seqence of distinct collision eents to occr. ith k distinct lineages and ait ntil to of them collide. This happens ith probability k(k 1) in each generation. Once a collision happens, e hae k 1 distinct lineages, and e 2N ait for to of them to collide ith probability (k 1)(k 2) per generation. Things contine 2N this ay ntil e are don to to distinct lineages, at hich point e ait for them to collide ith probability 2 = 1 per generation. The oerall process is shon, for or example ith 2N N k = 6, in Figre Gided by this ie of the process, e can analyze it as follos. Let W be a random ariable eqal to the nmber of generations back ntil coalescence. We can rite W = W k + W k 1 + W k W 2, here W j is a random ariable eqal to the nmber of generations dring hich there are

40 684 CHAPTER 21. EPIDEMICS exactly j distinct lineages. By linearity of expectation, e hae E [W ]=E [W k ]+E [W k 1 ]+ + E [W 2 ]. So it remains to figre ot the terms on the right-hand side. Each random ariable of the form W j can be ieed in the folloing ay: hen e hae j distinct lineages, e moe backard throgh sccessie generations, aiting ntil a particlar eent (a collision) first happens. We no make se of or approximations: W j is ery close to a simpler random ariable in hich e hae j lineages, e cont the nmber of steps ntil this nmber of lineages is redced to j 1, and in each generation this redction of lineages occrs ith probability exactly eqal to p = j(j 1). We let X 2N j denote this closely related, simpler random ariable; e rite X = X k + X k 1 + X k X 2, and e ill be interested in determining the expectation E [X] =E [X k ]+E [X k 1 ]+ + E [X 2 ] rather than the expectation E [W ]. Ho shold e think abot the expectation of one of these simpler random ariables X j? It is precisely as thogh e hae a coin that comes p heads ith a fixed probability p = j(j 1) per flip, and e ant to kno the expected nmber of flips ntil e see the first 2N heads. To compte this expectation, e recall Eqation (21.5) from earlier in this section, applied to the crrent random ariable X j : E [X j ] = Pr [X j 1] + Pr [X j 2] + Pr [X j 3] + The probability that X j is at least some ale i is jst the probability that the coin comes p tails on its first i flips, hich is (1 p) i. Therefore, E [X j ] = 1 + (1 p) + (1 p) 2 + (1 p) 3 + = 1 1 (1 p) = 1 p. This is a ery intitie relationship: the expected time to see the first heads on a coin ith a heads probability of p is simply 1 p. and so The random ariable X j describes precisely this process ith p = j(j 1). Therefore, 2N E [X] = 2N N N = 2N E [X j ]= 2N j(j 1), N j(j 1) + + 2N k(k 1) ( j(j 1) k(k 1) (21.8) ). (21.9)

41 21.9. EXERCISES 685 This last sm can be ealated by noticing that 1 j(j 1) = 1 j 1 1 j and applying this identity to each term in (21.9) e get ([ 1 E [X] =2N 1 1 ] [ ] [ j 1 1 ] [ 1 + j j 1 ] [ j +1 k 1 1 ]). k In this ne ay of riting the sm, almost all the terms inside the parentheses cancel each other ot, and the only to that srie are 1 and 1. Hence e conclde that k ( E [X] = 2N 1 1 ). k This gies s the reslt e ere looking for the approximate nmber of generations ntil coalescence and so e can conclde ith jst a fe final obserations. First, once k becomes moderately large, the expected time to coalescence depends only ery eakly on k; it is roghly 2N as k gros. Second, the breakdon of X into X k + X k X 2 lets s appreciate here most of the time is being spent in the merging to a common ancestor. As e begin moing backard in time, collisions happen relatiely qickly at first; bt as e contine moing backard, e find that essentially half the expected time is being spent once the lineages hae merged don to jst to, and these to are searching for a final collision point at the most recent common ancestor. Third, the approximations proide s ith a ery simple ay to bild trees according to this process: e simply follo the recipe in Figre 21.21, draing parallel lines backard in time ntil the otcome of a coin-flip tells s to pick to of the lines niformly at random and merge them into one. Finally, e shold note that althogh e hae introdced some approximations into the original formlation of the problem, sbseqent ork has shon that the final estimates are ery close to the exact reslts one gets throgh mch more intricate analysis [91, 174] Exercises 1. Sppose yo are stdying the spread of a rare disease among the set of people pictred in Figre The contacts among these people are as depicted in the netork in the figre, ith a time interal on each edge shoing hen the period of contact occrred. We assme that the period of obseration rns from time 0 to time 20. (a) Sppose that s is the only indiidal ho had the disease at time 0. Which nodes cold potentially hae acqired the disease by the end of the obseration period, at time 20?

42 686 CHAPTER 21. EPIDEMICS x [4,8] [1,3] [14,18] s [5,9] y [7,12] [10,16] [12,16] z Figre 21.22: Contacts among a set of people, ith time interals shoing hen the contacts occrred. (b) Sppose that yo find, in fact, that all nodes hae the disease at time 20. Yo re fairly certain that the disease coldn t hae been introdced into this grop from other sorces, and so yo sspect instead that a ale yo re sing as the start or end of one of the time interals is incorrect. Can yo find a single nmber, designating the start or end of one of the time interals, that yo cold change so that in the reslting netork, it s possible for the disease to hae floed from s to eery other node? a b c d e Figre 21.23: A contact graph on fie people. 2. Imagine that yo kno a contact graph on a set of people, bt yo don t kno exactly the times dring hich contacts happened. Sppose yo hae a hypothesis that a particlar disease passed beteen certain pairs of people, bt not beteen certain other pairs. (Let s call the first set of pairs positie, and the second set of pairs negatie.) It s natral to ask hether it s possible to find a set of time interals for the edges that spport this hypothesis in a strong sense: they make it possible for the disease to flo beteen the positie pairs, bt not beteen the negatie pairs. Let s try this genre of qestion ot on the simple contact graph among fie people shon in Figre

43 21.9. EXERCISES 687 (a) Can yo find time interals for the edges that make it possible for the disease to flo from eery node to eery other node, ith the one exception that it is not possible for it to flo from node a to node e? If yo think it is possible, describe sch a set of time interals; if yo think it is not possible, explain hy no sch set of time interals exists. (b) Can yo find time interals for the edges that make it possible for the disease to flo from a to d and from b to e, bt not from a to c? Again, if yo think it is possible, describe sch a set of time interals; if yo think it is not possible, explain hy no sch set of time interals exists. 3. Imagine that yo re adising a grop of agricltral officials ho are inestigating measres to control the otbreak of an epidemic in its early stages ithin a liestock poplation. On short notice, they are able to try controlling the extent to hich the animals come in contact ith each other, and they are also able to introdce higher leels of sanitization to redce the probability that one animal passes the disease to another. Both of these measres cost money, and the estimates of the costs are as follos. If the officials spend x dollars controlling the extent to hich animals come into contact ith each other, then they expect each animal to come into contact ith 40 x 200, 000 others. If the officials spend y dollars introdcing sanitization measres to redce the probability of transmission, then they expect the probability an infected animal passes it to another animal contact to be.04 y 100, 000, 000. The officials hae to million dollars bdgeted for this actiity. Their crrent plan is to spend one million on each of the to kinds of measres. Using hat yo kno abot epidemics, old yo adise them that this is a good se of the aailable money? If so, hy? If not, can yo sggest a better ay to allocate the money?