key: cord-0034642-nhw3tome
authors: Arinaminpathy, Nimalan; Metcalf, Charlotte Jessica E.; Grenfell, Bryan T.
title: Viral Dynamics and Mathematical Models
date: 2014-02-27
journal: Viral Infections of Humans
DOI: 10.1007/978-1-4899-7448-8_5
sha: c6a033d784a9be08b8dda15cdf0f35f17e86effb
doc_id: 34642
cord_uid: nhw3tome

Mathematical tools have been widely applied in understanding the dynamics and control of viral infections. Here we present some fundamental aspects of infection dynamics, starting with acute immunising infections as a case study for herd immunity and other important factors in the spread and control of infection. We then discuss the dynamics of infections with more complex life histories, including chronic infections, and those showing evolution for immune escape. We conclude with a discussion of important gaps in our current understanding of viral dynamics, along with future research needs.

The application of mathematics to explore the dynamics and control of viral infections has a long history, which predates its application in many other fi elds of biology, as well as the germ theory itself [ 1 ] . A particularly fruitful area for development has been the dynamics of acute immunizing infections, with their relatively simple natural history and rich notifi cation time series [ 2 -4 ] . The last three decades have seen a considerable upsurge in the use of mathematical and computational models to explore the dynamics and control of a wide range of viral infections. This phase arose initially from developments in ecological population dynamics [ 4 ] ; it was then greatly accelerated, both by the explosion in computational power and by the emergence of human immunodefi ciency virus (HIV) infection, severe acute respiratory syndrome (SARS), and other potential pandemic threats [ 5 ] .

In this chapter, we review basic concepts in infection dynamics and control, via a synthesis of epidemic models and data. We begin with acute immunizing infections, focusing on measles as a case study of the impact of herd immunity and other determinants of epidemic dynamics. We then extend the discussion to dynamical modeling applications to a range of other acute and chronic viruses, with a variety of more complex life histories. After a brief discussion of models for the within-host dynamics of viruses, we conclude with suggestions for gaps in knowledge and future research needs.

Dynamics of Acute Immunizing Infections

Acute immunizing infections typically generate recurrent epidemics in large communities. We illustrate these patterns with the best documented case -the dynamics of measles in large cities in England and Wales [ 2 , 3 ] . Figure 5 .1a shows the time series of raw weekly notifi cations for measles in London from 1944 (shortly after measles cases became notifi able) to 1994. These data indicate three fairly distinctive dynamical eras: • 1944-1950: Principally annual epidemics. • 1950s-1960s: Regular, mainly biennial epidemic cycles, with intervening small annual peaks -incidence rates are markedly seasonal (Sect. 2.4 ). • 1970s onwards: The vaccine era brought declining incidence, with increasing irregular, lower-amplitude epidemic cycles. By the end of the series shown in this fi gure, cases became very sporadic, with increasing levels of mis-notifi cation of clinically identifi ed cases [ 7 ] . As well as these fl uctuations in individual large cities, there are also rich dynamical patterns in the spatial spread of epidemics among large and small communities (Sect. 4 ). Regional and temporal demographic variations, especially in birth rate , can also markedly infl uence dynamics (Sect. 2.4 ).

The striking epidemic patterns of acute immunizing infections are particularly well documented for widely notifi able infections such as measles. The process of explaining these cycles, as well as that of addressing associated public health issues, has spawned an extensive analytical literature, spanning public health epidemiology, theoretical biology, and population dynamics [ 4 , 5 , 8 -10 ] . The key conceptual tool has been a family of compartmental dynamical models, based on the SIR (susceptible-infected-recovered) paradigm [ 2 -5 , 11 , 12 ] . As described below, the SIR family successfully captures many key features of the epidemiological dynamics and control of viral infections. The basic SIR formulation embodies the dynamics of immunizing infections. Because viruses reproduce rapidly in the host, we ignore within-host kinetics in the simplest SIR models (but see Sect. 8 ); depending on model details, this leads to a compartmentalization of the host population between (as yet uninfected) susceptible individuals, infected, recovered, and other classes. An initial taxonomic split here is between the SIR model (which crudely assumes that all infected individuals can pass on infection) and the SEIR (susceptible, exposed, infected, recovered) model, which adds an "exposed" (infected but not yet infectious) class [ 4 ] . SIR and SEIR models have qualitatively very similar dynamics [ 13 ] ; however, we describe the latter with its more realistic depiction of viral incubation. The basic SEIR model, illustrated in Fig. 5 .1b , refl ects the following set of biological assumptions; we use measles as the classic acute exemplar here [ 14 ] . After a few months of maternally derived passive immunity, infants enter the virus-naïve "susceptible" (S) class; susceptibles can then become infected by close contact with infectious individuals (generally via respiratory aerosol for measles). Infection moves individuals into the "exposed" (E) class, where they incubate but do not transmit the infection for around a week; this leads to the infectious (I) class, where virus is shed, again for approximately a week, after which individuals enter the recovered state (R). In the basic SEIR model, recovered individuals are assumed to be immune for life, both to clinical disease and to retransmitting the infection. However, subclinical infection (and hence boosting of immunity [ 15 , 16 ] ) is in principle possible. The lifelong sterilizing immunity induced by such an infection also makes this an excellent potential vaccine candidate [ 4 ] . For the SEIR model, vaccination is at its simplest assumed to be delivered to a proportion p of infants at the end of the maternal immunity period (i.e., near the effective "birth" of susceptibles). For individuals who seroconvert, immunity is then often assumed to be lifelong, moving susceptibles into the recovered class ( Fig. 5.1c ) . There is also a considerable literature exploring more operationally realistic age distributions and immunogenic characteristics of vaccines in specifi c cases [ 5 , 17 ] .

We can lay bare much of the dynamical behavior of immunizing epidemics by considering the case of the "simple" epidemic ( Fig. 5 .1c , d ), in which the outbreak occurs suffi ciently rapidly to ignore processes of host birth and death. The talismanic quantity here is the basic reproduction number (or ratio) of infection, R 0 [ 4 , 5 ] . At its simplest, R 0 is defi ned as the total number of secondary cases caused by an infectious individual when introduced into a well-mixed fully susceptible population. To illustrate the ensuing dynamics, consider an epidemic of infection with R 0 > 0. Since each case initially causes R 0 secondary infections, the epidemic increases more or less exponentially over the fi rst few infectious generations ( Fig. 5.1c ). However, these dynamics rapidly deplete the susceptible pool. This brings us to the other key parameter of epidemic spread: the effective reproduction number, defi ned as R = [ S / N ] R 0 . Here, S / N is the proportion of susceptibles in the population and R is the realized value of R 0 as the epidemic develops [ 4 ] . Since S declines over the course of the epidemic (Fig. 5.1c ), so does R (Fig. 5.1d ). This progressive decline in secondary infection rates through the epidemic is a manifestation of the key epidemiological process of herd immunity [ 18 ] -increasing natural immunity of the population that indirectly protects the remaining susceptibles from infection. Eventually, the effective reproduction ratio declines through unity (Fig. 5 .1c ) as population immunity increases; this corresponds to the herd immunity threshold, above which the epidemic will always decline, even if reintroduced to a closed population. This threshold is thus also a key aim of vaccination campaigns [ 4 , 5 ] . Remembering that R = [ S / N ] R 0 , the associated susceptible proportion at the herd immunity threshold ( R = 1) becomes s c = 1/ R 0 ( Fig. 5 .1c ); thus, immunizing at a level above p c = 1 − s c = 1 − 1/ R 0 will eliminate local transmission. As described below, these calculations have been refi ned considerably to allow for heterogeneities in transmission with age, space, and other characteristics (Sects. 3 , 4 , and 5 ). Nonetheless, our simple expression for p c is an extremely useful metaphor for epidemic control: (1) because of indirect protection, not everyone needs to be vaccinated to eliminate transmission (corresponding to p c < 1) and (2) the effort required to increase this level of immunization increases with transmission rates. The latter point is made clear by a simple plot of p c against R 0 ( Fig. 5 .2 ); more transmissible immunizing infections such as measles are harder to control than less transmissible agents such as smallpox. However, this refers to "random" mass vaccination; more targeted strategies such as ring vaccination coupled to active surveillance can promote elimination even below p c , given (as with smallpox elimination [ 19 ] ) the right biological characteristics and logistics. Partial immunity and other characteristics of the population can complicate the picture still further (see Sect. 5.2 below). Nonetheless, the metaphor that more transmissibility necessitates a stronger vaccination effort remains.

In the simplest analysis of an immunizing infection (Sect. 2.2 ), sustained cycles of infection will disappear, and the infected proportion in the population will settle to a constant level The question of what maintains the recurrent epidemics [ 9 ] generally observed for immunizing childhood infections (e.g., Fig. 5 .1 ) drove researchers to seek the key aspect of biological realism missing from the simplest SEIR model. For measles in England and Wales, seasonal variation in transmission driven by increased contact rates of children when schools were in session rapidly emerged as a possible candidate [ 12 , 21 ] and has since received considerable empirical support [ 2 , 3 ] . Stochastic fl uctuations in incidence could in principle also contribute to the maintenance of cycles [ 22 , 23 ] ; however, for measles, the predominant driver is seasonality in transmission [ 2 , 3 ] . Birth rate may modulate the periodicity of recurrent cycles driven by seasonality through its role in determining the rate of susceptible replenishment [ 24 ] . For example, for most of the postwar pre-vaccination era in London, seasonality generated sustained cycles by resonating with the biennial epidemic tendency of measles dynamics. However, during the postwar baby boom, births achieved suffi cient levels to shift measles dynamics into annual cycles ( Fig. 5.4 ).

It was realized early that seasonality could also result in more complex dynamics, including chaotic fl uctuations, that is, very irregular dynamics with little long-term predictability. Contexts with both high and low birth rates (and both high and low transmission since birth rates and transmission act dynamically in very similar ways for immunizing infections [ 24 ] ) can promote complex dynamics via coexisting attractors. Until recently, it was thought that chaotic measles dynamics were not likely to [ 20 ] ). The green trajectory shows the joint dynamics of susceptible and infectious densities through time; area plots show the dynamics of infectious and susceptible individuals (Taken from Fig. 3 in above book chapter, found on p. 39) be observed [ 3 , 25 ] , since observed seasonality in transmission was not strong enough to drive the associated violent epidemics. However, recent analyses of measles in Niger revealed very strong seasonality (driven by movement in and out of cities linked to rainfall); this, combined with high transmission rates and the highest birth rate in the world, results in irregular outbreaks consistent with expectations of chaos ( Fig. 5 .5 ). Erratic boom and bust outbreaks are expected to continue even as routine vaccination improves; and this suggests that high investment in reactive vaccination and surveillance is important, and pulsed vaccination approaches such as supplementary immunization activities could also play a helpful role in synchronizing dynamics [ 26 ] . Age Structure, Demography, and Serological Profi les

The mean age at which individuals become infected by immunizing viral infection is generally lower for infections with higher rates of transmission or R 0 (defi ned above). Intuitively, this occurs because the faster an immunizing virus is spreading through a population, the younger the age at which individuals are likely to be exposed to it. Host demography modulates this relationship, and higher birth rate countries with an R 0 equivalent to that in lower birth rate countries will tend to have a lower average age of infection [ 27 ] . Maternal immunity, or protection of children after birth by transfer of maternal antibodies, will also have an impact, increasing the average age of infection for children whose mothers have been exposed to infection. Since maternal antibodies rarely persist for much more than a year, this effect will be greatest for infections with very low average ages of infection [ 4 ] . Beyond these broad descriptors, however, there is the added complication of possible relationships between age and probability of exposure to infection. Such relationships may arise for a variety of reasons. For example, only individuals beyond a certain age may work in areas where the disease is transmitted, or the disease may be only sexually transmitted. As a result, the force of infection, or probability that a susceptible individual will be infected, will show a distinct relationship with age. For directly transmitted infections, like measles, mumps, rubella, and infl uenza, the age profi le of the force of infection is determined by the rate at which individuals of different ages interact. There are two approaches to estimating this variation. The fi rst is to use age profi les of seropositivity to infer the pattern of the force of infection and, from this, extrapolate to the pattern of contacts over age [ 28 -31 ] . The link between these age profi les and the force of infection over age can be made since once an individual has been infected, she/he can never be infected again. Therefore, to be immune at any age, the individual must have contracted the infection prior to that age. On a population level, crosssectional seropositivity thus provides an indication of the total risk of infection for individuals up to a given age.

(Longitudinal age-serological profi les can be even more powerful in quantifying epidemic risk [ 32 ] ). The age profi le of infection can be framed mathematically [ 33 ] , and parameters describing contacts between individuals of different ages inferred [ 31 ] . The resulting estimated matrix of contacts over age is known as the Who-Acquires-Infectionfrom-Whom or WAIFW matrix. Alternatively, patterns of contacts between individuals can be directly measured [ 34 ] , for example, using diary studies (Fig. 5.6 ) ; and by combining this with the age profi le of infection within the population, the force of infection over age can be inferred [ 35 ] .

The infection dynamics themselves may also infl uence the relationship between age and force of infection. If outbreaks are separated by long intervals during which little exposure occurs, individuals may remain susceptible for many years and thus contract the infection at a later age than might be predicted from the age profi le of contacts alone [ 36 ] . More subtly, detailed network analyses show that even within a single infl uenza outbreak, the burden of disease can cascade from children (where contacts are highest) to the less connected adults as immunity accumulates within the children, with implications for optimal vaccination distribution [ 37 ] .

Understanding the processes underlying the average age of infection has a practical importance for any infection where the burden of disease shows an age profi le. Rubella is a classic example. Infection during early childhood tends to be mild, but infection during pregnancy may result in birth of a child with congenital rubella syndrome, consisting of a range of birth defects (see Ref. [ 38 ] ). A realistically complex age structure of mixing, as detailed in empirical studies, may thus be of crucial importance in establishing the burden of disease [ 39 ] but also how the burden of disease is likely to change as a result of vaccination.

So far, we have assumed that deterministic, spatially homogeneous dynamics govern infectious disease outbreaks. In fact, epidemics often spread across a heterogeneous landscape of human cities, towns, or rural communities, and this spread depends partly on the links between those locations. This leads us to move from the deterministic SIR model described above to stochastic models, which account for the random nature of individual infection dynamics and demography -for instance, individuals may or may not become infected with a given average probability, so that by chance, particularly in small populations, no new infections may occur, and the chain of transmission may be broken. During the troughs between epidemic outbreaks in smaller communities, incidence may fall to such low numbers that local extinction is likely. Based on this observation, Bartlett [ 9 , 40 ] used analyses of epidemiological data and stochastic models to develop the notion of critical community size (CCS), or population size below which stochastic extinction is expected, which was further developed by Black [ 41 ] in studies of measles persistence in insular populations. Bartlett demonstrated the existence of a CCS of around 300,000-500,000 for measles in England and Wales. For measles to persist in locations with a population size smaller than this CCS, immigration of infected immigrants from elsewhere in the metapopulation is necessary. The result is that the spread of measles across England and Wales in the pre-vaccine era resembled traveling waves spreading out from London [ 6 ] , with a substantial epidemic lag in locations further away. The duration of the lag was also shaped by the size of the local populations. The duration of fade-outs following local extinction contains information on the degree to which a particular location is connected to the metapopulation as a whole [ 42 ] . Generally, this points to larger places being more connected. More detailed parametric analyses tend to confi rm this (e.g., the gravity model [ 43 ] ).

Similar processes may operate for immunizing or lethal infections in animal populations, with, for example, the spread of rabies through raccoon populations acting as an invasion wave structured by large landscape features such as rivers [ 44 ] . For more complex human infections, features such as imperfect immunity (see Sect. 5.2 below) will tend to shift the age class of hosts who disperse the infection. That shift may impact on the main mechanism of dispersal [ 45 ] and move the scenario away from invasion into a locally coupled landscape to a more demographically driven dynamic [ 46 ] .

So far, we have explored the epidemiological dynamics of acute, immunizing viral infections. Though the resulting dynamics are both important and fascinating from a dynamical perspective, the natural (life) history of most viral infections departs, in one way or another, from this simple case. We review these complexities, and their epidemiological and control implications, in succeeding Sections.

A dramatically different life history from the transient infection paradigm represented by measles is observed with infections that are much more persistent (even lifelong). This difference may be expressed in terms of infectious period in individual hosts [ 4 ] . To illustrate how increasing infectious period alone modifi es the violent epidemics of childhood infections, we retain the assumption of lifelong immunity of the SEIR model and vary the infectious period, from the roughly 1 week of measles to 10 years, corresponding to the approximate average pre-HAART treatment infectious period of HIV [ 4 ] . Figure 5 .7 explores this comparison, simulating an infection which invades a partly susceptible population (full details are given in Ref. [ 6 ] ; note that for clarity, the epidemic curves for 1-and 10-year infectious periods are raised above the curves for the more acute infections). Each simulation refers to a virus with a different infectious period. We assume that R 0 is identical for each of these infections. R 0 is roughly given by the product of mean per capita infection rate and infectious period; thus, to keep R 0 constant: • A short infectious period implies a relatively high infection rate. • A longer infectious period requires a lower infection rate.

This assumption imposes a simple evolutionary constraint on our set of model pathogens in Fig. 5 .7 , in that fi tness (roughly equating to R 0 ) is kept constant as we increase the infectious period. Figure 5 .7 indicates, fi rst, that increasing the infectious period reduces, and eventually eliminates, the tendency for cyclical epidemics [ 48 ] ; essentially, a longer infectious period "fi lls in the troughs" following major epidemics. As infectious period increases, we therefore see a transition from seasonally driven biennial epidemics (at the measles extreme of a 1-week infectious period) to lowamplitude annual epidemics (at 1 month). For longer infectious periods (1 year), we see slowly evolving epidemics with little seasonal activity and a modest post-epidemic overshoot. Finally, for a 10-year infectious period, we see a smooth slow epidemic, with an essentially "logistic" rise to a stable endemic plateau of incidence. Though crude, this exercise captures the essential dynamical transmission from the violent epidemics of acute infections to the much smoother and slower epidemic invasion of HIV [ 4 ] . Note that the seasonal variation in infection rate is assumed similar for the "measles" and "HIV" cases; however, the latter completely eliminates associated seasonal fl uctuations in incidence due to the smoothing effect of prolonged infectious carriage. Figure 5 .7 also illustrates a second major dynamical impact arising from the trade-off of increased infectious period against lowered infection rate. "Fast" infections are much more prey to local stochastic extinction in the deep troughs between epidemics (Sect. 2.4 ) than the much more endemic incidence promoted by longer infectious period. On the other hand, acute immunizing infections can invade populations much more quickly than chronic infections for the same R 0 (Note that very imperfect immunity (corresponding to 'SIS' dynamics) could generate 'fast' invasion even for relatively chronic infections ( Figure 5.7 , inset) , because of the increased supply of susceptible individuals (Sect. 5.2 )). Note that, despite these great variations in dynamics, the assumption of a common R 0 means that the herd immunity threshold for elimination of infection is the same across this range of behaviors Sect. ( 2.3 ).

We have seen that the SEIR framework ( Fig. 5.1b ) can be a powerful tool for capturing the dynamics of perfectly immunizing infections such as measles. With its various refi nements to incorporate structure in the host population (such as age and spatial distribution), it yields a rich variety of dynamical behavior (Fig. 5.7 ) . Given the range of immunological complexities and different viral life histories, however, it is clear that SEIR dynamics are only part of the story. In such cases, it is often possible to adapt Fig. 5 .1b to more faithfully refl ect such important biological complexities.

Many viruses are capable of reinfecting a host. This may be due to the waning of host immunity over time, as with respiratory syncytial virus (RSV) [ 49 ] , or due to viral evolution for immune escape (e.g., with norovirus [ 50 ] and infl uenza) [ 51 , 52 ] : see Sect. 6 for a more in-depth treatment of the latter. In either case, the overall dynamical effect is for recovered, immune individuals to eventually reenter the susceptible pool, as illustrated by Fig. 5.8a . Where such waning of immunity is appreciable, the implications for vaccination can be profound. As the susceptible pool is replenished not only by births but also by previ-ously immune individuals, the critical vaccination threshold (Sect. 2.3 ) is raised, making it more diffi cult to control the spread of infection. For this reason, vaccination against many imperfectly immunizing infections is aimed at direct protection of those receiving vaccine, rather than at raising indirect protection.

So far, the models we have considered assume that immune individuals have solid, transmission-blocking immunity, even if this should wane through time. In reality, however, it is common for immunity to be clinically protective (i.e., against symptomatic disease) but only partially protective from infection. This applies, for example, in the case of rotavirus, the leading cause of severe diarrhea in infants worldwide [ 53 , 54 ] . Figure 5 .8b illustrates a model structure developed to capture these dynamics [ 46 ] .

Recalling the qualitative differences between the dynamics of acute and chronic infections (Fig. 5.7 ) , some viruses show a "mixed" character on the host population level: while some individuals clear infection relatively quickly, others may continue to harbor the virus as "carriers," either in a latent phase or in a chronic infectious state, for a longer period. An example is hepatitis B, in which 5-10 % of adult patients show chronic infection. As illustrated in Fig. 5 .8c , this can be represented mathematically by identifying two infectious classes: one acute and the other chronic. Such factors can have implications for the dynamics and persistence of infections in a population [ 55 ] . In particular, where a disease shows epidemic cycles, facing the possibility of local extinction in the epidemic troughs, the presence of a small number of carrier hosts can facilitate viral persistence in the population, through these troughs. For example, in some cases infection with varicella zoster virus can be followed decades later by infectious shingles, an occurrence which could maintain the virus in small populations [ 56 ] .

Strain structure, vector dynamics, and seasonality in transmission offer yet more complexities, as in the example of dengue [ 57 ] . Spread by Aedes mosquitoes, dengue has shown a signifi cant emergence worldwide in the last 50 years.

Dengue dynamics are notable for the global circulation of four distinct serotypes (see Fig. 5.9a ) . Notably, prior immunity to a given serotype can elevate the risk of severe disease from subsequent infection by a different serotype [ 60 -62 ] . It has been proposed that this arises from antibody-dependent enhancement (ADE) between different serotypes [ 63 , 64 ] , and modeling studies suggest that ADE could play an important role in shaping the epidemiological and strain dynamics of dengue, on the host population level [ 65 -67 ] .

However, there are also indications of a short period of cross-protective immunity (i.e., against all serotypes) for 2-9 months following infection [ 68 ] , potentially mediating some degree of competition between serotypes. Several studies have addressed the dynamical implications of interactions between ADE and such immunity [ 69 , 70 ] . Dengue dynamics are therefore complex and multifactorial: an understanding of these effects is important in understanding the potential for unexpected effects from transmission-reducing interventions [ 71 ] .

Had viruses been discovered by the time that Darwin proposed "descent with modifi cation" in the mid-nineteenth century, they may quickly have been recognized as fi ne examples of evolution in action. Indeed, today there is wide acknowledgment of the inextricable role that evolution plays in viral dynamics [ 58 ] and in the control of many different viral infections. RNA viruses in particular, lacking the replication fi delity of a DNA genome, are capable of considerable mutation rates [ 72 ] . In the presence of host immunity, natural selection

Examples for extensions of the basic SEIR model (Fig. 5.1b ) .

( a ) Waning immunity can be represented by allowing individuals to return from recovered ( R ) to susceptible ( S ) status at a given rate. ( b ) If acquired immunity affords clinical but not sterilizing (transmission-blocking) pro-tection, then additional SEIR stages may be incorporated to capture the corresponding effects on transmission dynamics. ( c ) An example of population heterogeneity, distinguishing "acute" cases ( I A ) from "carriers" ( I C ), the latter recovering at a slower rate than the former will favor those mutants most capable of evading immunity while still being capable of spreading between hosts. As a result of these high mutation rates, population-level evolutionary patterns that can arise are quite diverse. Figure 5 .9a illustrates how some infections, such as HIV, show signifi cant population-level diversity [ 73 ] , while others, notably seasonal infl uenza A, exhibit a markedly different pattern, with a "trunk-like" phylogeny on the population level [ 74 ] .

The case study of infl uenza demonstrates not only interesting evolutionary patterns but also the following important features:

(i) The challenges that evolution can pose for infection control (ii) The complex interplay, across a range of physical scales, between the evolution of a pathogen and its success in spreading in a host population (iii) The many outstanding questions that remain, in understanding how viruses adapt to continue reinfecting their hosts

In what follows, we explore both within-host and population-level aspects of infl uenza evolution, with an emphasis on seasonal (interpandemic) infl uenza. We then set out the prevailing paradigms that seek to explain how these patterns arise, through complex interactions between viral evolution and epidemiology.

Classic theoretical principles [ 4 ] provide a useful framework in which to think about viral evolution, where "evolutionary fi tness" is determined ultimately by the capacity for transmission in a given host population. Recall from Sect. 2.3 that in the presence of prior immunity in the population, the effective reproduction number, R , serves as a compact measure of transmission potential. Vaccination aims to lower R through reducing the number susceptible in the population; however, evolving pathogens introduce the complexity of countering this effect, by evading immunity and thus acting to restore R . Note the combination of immune escape and transmissibility encapsulated in R . In particular, on the scale of the host population and with a given level of existing immunity, the escape mutant with the greatest evolutionary fi tness is that which maximizes R [ 5 ] .

As discussed in Chap. 21 , the infl uenza surface protein hemagglutinin (HA) is a key immune target, with anti-HA antibodies capable of offering sterilizing immunity, that is, the potential to block host infection altogether. Indeed, raising such immunity is a central function of current infl uenza vaccines. However, HA is also the most variable viral component, continually under selective pressure to escape antibody binding [ 51 ] . For these reasons, while infl uenza evolution is by no means limited to HA, this viral component has attracted the most attention. An important mechanism of immune escape is through conformational changes of HA epitopes to abrogate antibody binding, without compromising the ability of the virus to attach to host cells [ 75 ] . More recently, however, results from mouse experiments suggest that escape mutants can also acquire increased viral avidity for host cells [ 76 ] . Another potential immune escape strategy [ 77 ] is glycosylation, or the attachment of oligosaccharides to the HA molecule, to occlude epitopes from antibody binding. Such strategies are not without potential functional costs for replication [ 78 ] , and yet there are indications that human infl uenza A/H3N2 has been accumulating glycosylation since its introduction into the human population in 1968 [ 79 , 80 ] .

On a genetic level, the evolutionary pattern for infl uenza HA ( Fig. 5.9a ) , showing serial replacement of strains through time, is often characterized as "drift." The limited standing diversity, or "trunk-like" phylogeny, is especially paradoxical in light of the relatively high mutation rate of infl uenza A, where instead several lineages might have been expected to emerge and coexist in the population. Moreover, how do such genetic patterns translate to antigenic properties, that is, viral interaction with anti-HA antibodies? In 2004, new techniques allowed a characterization of the antigenic evolution of the infl uenza subtype H3N2 [ 74 ] during the years since its introduction into humans in 1968. The resulting pattern is characterized by "punctuations" in antigenic evolution, occurring roughly every 2-8 years, in contrast with the more gradual pattern of genetic change shown in Fig. 5 .9a . These punctuations have great importance for public health, often necessitating major reformulation of current, HA-based vaccines [ 81 ] .

Two prevailing paradigms explaining these important phenomena, on both the genetic and antigenic levels, are the epochal evolution model [ 82 ] and that of strain-transcending immunity [ 83 ] . Both may be considered "phylodynamic" models [ 58 ] , in the sense of aiming to capture the complex interactions between viral immune escape, viral population genetics, and epidemiology.

The picture of "strain-transcending immunity" [ 83 ] invokes a temporary mode of immunity, which immediately follows recovery from infection and protects the individual against infection by all infl uenza strains. Such broad immunity is postulated to arise, for example, either from T cells or from innate immunity [ 83 ] . Over the course of months, this broad immunity gives way to more long-lived, more narrowly acting immunity that is specifi c to the immunizing strain. On the population level, a key dynamical effect of such immunity structure is to impose population-level constraints on viral diversity, suffi ciently strong to maintain a single dominant lineage through time.

The epochal evolution model [ 82 ] provides an alternative view. It builds on the proposal by Kimura in 1968 [ 84 ] that many amino acid substitutions in nature do not alter evolutionary fi tness and are thus phenotypically neutral . If infl uenza HA evolution can be thought of as tracing a series of paths through a space of genotypes, the epochal evolution model proposes that, in phenotypic terms, such a space has a modular structure (Fig. 5.9b ) in which each "module" is a group of genotypes sharing the same antigenic phenotype and a transition between modules corresponds to the observed antigenic "jumps." In particular, HA evolution diffuses through the local genotype space, accumulating neutral substitutions and thus genotypic diversity, over several years. Ultimately, a single strain accumulates the substitutions required to transition to a new antigenic phenotype. At this point, the emergent strain -owing to its antigenic novelty -causes a peak in infection and undergoes a selective sweep in the host population, to the exclusion of other strains. Such an event thus acts to periodically control antigenic diversity, maintaining the "trunk-like" shape of the infl uenza phylogeny.

Alongside these two prevailing paradigms of infl uenza evolution, yet another model [ 85 ] proposes that observed strains are drawn from a limited set of antigenic types, with their selection dependent on niches in host population immunity. While each of these models succeeds in capturing important features of infl uenza dynamics, they also highlight important gaps in our understanding of viral evolution and how to manage it.

The degree to which viruses harm hosts varies considerably. Both within and between virus species, some variants may barely affect hosts at all, while others have signifi cant health impacts, up to and including host mortality [ 86 ] . An evolutionary perspective indicates that variation in virulence across clones should be shaped by the costs and benefi ts of virulence for parasite transmission [ 87 ] . The spread and persistence of virulence mutations may be favored if they covary with transmission. However, since host mortality can interrupt transmission, a range of values of virulence may lead to equivalent levels of overall fi tness for the parasite [ 88 ] . While it is broadly agreed that the theoretical framework and empirical evidence on the existence of virulence-transmission trade-offs need to be further developed and extended [ 87 ] , there are some systems where the general predictions appear to be borne out (e.g., [ 89 , 90 ] ). Perhaps most famously, the introduction of the myxoma virus into Australia was at fi rst devastating to rabbit populations, causing near 100 % mortality. However, over subsequent years, the virulence of the infection decreased. Various lines of evidence suggest that this occurred in response to selection for increased transmission via decreased virulence -swift rabbit mortality was not an effi cient mode of transmission for the virus.

While we have so far largely discussed virus dynamics on the level of the host population, there are equally important processes to be considered on the within-host level. The "kinetics" of viral infection arise from a complex array of factors including viral replication and variability, the dynamics of the immune response, and pathogenicity to the host [ 91 ] . Over the past two decades, mathematical models have played an important role in studying these dynamics, often motivated by the need to understand the actual and potential impact of interventions such as treatment or immunization [ 91 -93 ] . A major example is HIV infection (see Chap. 43 ); its clinical course is characterized by an initial acute phase of viral replication, lasting on the order of weeks, followed by an asymptomatic latent phase that can last decades before ultimately progressing to AIDS [ 94 ] . Major advances in the 1990s showed that despite the long clinical timescale involved, in vivo HIV replication is in fact a rapid process, with the viral life cycle being on the order of days [ 95 , 96 ] . Subsequent studies modeling the dynamics of drug resistance underlined the need for early and aggressive drug therapy [ 97 ] . Conversely, more recent work capturing the dynamics of the immune response has illustrated how immunity raised by CD8+ T-cell vaccination elicits a response that is too weak and too late to achieve sterilizing immunity [ 98 ] .

Hepatitis C virus (HCV) represents another major public health challenge and, like HIV, is a rapidly mutating virus capable of continually evading immunity to establish chronic infection [ 99 ] . HCV infection is currently treated with broad-spectrum combination antiviral therapy including interferon-alpha, ribavirin, and protease inhibitors, with upwards of 50 % of treated patients being responders; however, there is a need for a more rationally optimized approach. Models capturing viral dynamics in treated and untreated patients have contributed to an understanding of the action of these therapies [ 100 -102 ] , estimates of parameters such as rates of viral growth and of viral RNA clearance [ 102 , 103 ] , and correlates of long-term response to therapy [ 104 ] .

Modeling approaches have been used to study the kinetics of acute infections too, notably in the context of infl uenza infection [ 105 ] . A key interest, for example, has been the relative importance of target cell depletion, innate immunity, and adaptive immunity in shaping the dynamics of viral infection [ 106 -108 ] .

Overall, while there remain signifi cant gaps in our understanding of these and other major viral infections [ 103 , 109 ] , this work demonstrates the clinically relevant insights that can be derived from a careful study of within-host dynamics [ 91 , 110 ] .

This chapter has broadly outlined the multitude of factors shaping viral dynamics, as well as the role of mathematical approaches in elucidating these factors and their interactions. This growing body of work sets the stage for future directions. The link between epidemiological modeling and policy is long-standing. Bernoulli's work provides perhaps the earliest case study [ 1 ] ; recent high-profi le examples include the use of models to guide responses to foot and mouth disease in the UK [ 111 ] , smallpox [ 112 ] , and infl uenza [ 113 ] . The key criterion of the effective deployment of models for policy is that they are embedded in testable hypotheses, embodying the best possible science. Continuing progress in this area thus calls for better understanding of the basic principles.

Especially in the context of viral evolution, whether in relation to immune escape or virulence, a complete biological framework calls for a linkage of processes across disparate scales, from the macroscopic structure of the host population to the molecular basis of viral replication and transmission. A key task at the heart of this challenge is arguably to quantify transmission potential in terms of immune escape, that is, to measure empirically how changes at the molecular level impact R (see Sect. 6.1 ). Indeed, recent equine infl uenza experiments provide some important steps in this direction [ 114 ] .

New and emerging ways of tracking disease may also help to shed new light on the global spread of viruses and on how they may be better controlled. To name three examples, while disease surveillance continues to operate largely through public health channels, there is increasing interest in the use of alternative sources, including automated tracking of Internet and social media trends [ 115 , 116 ] . Second, genetic sequencing and analysis complements existing epidemiological approaches. The advent of highthroughput whole-genome sequencing is expected to shed new light on within-host viral diversity [ 117 ] , an important aspect in our understanding of viral evolution. Third, any viral infection leaves its mark on the host immune system. Serological studies thus offer another valuable source of information in monitoring diseases [ 32 ] and already play an important role in many national surveillance programs (see Chap. 4 ). While in some cases there remain challenges in the clinical interpretation of quantitative serological data (e.g., Ref. [ 118 ] ), its future use in estimating, for example, the prevalence of subclinical, but infectious, cases could be of signifi cant value to public health efforts [ 119 -121 ] .

Finally, it is important to consider the role of the hosts themselves. With the host population providing the medium through which viruses spread, important factors in viral dynamics include heterogeneities among individuals (e.g., host genetic variation [ 122 ] and "superspreaders" of infection [ 123 ] ) and patterns of human connectivity and mobility. Various approaches are beginning to unravel some of these patterns, both in the developed [ 34 , 124 ] and in the developing world [ 125 ] . Future developments in such techniques will provide valuable new data for understanding the human role in viral dynamics.

Essai d'une nouvelle analyse de la mortalité causée par la petite vérole et des advantages de l'inoculation pour la prévenir

Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR model

Dynamics of measles epidemics: scaling noise, determinism, and predictability with the TSIR model

Infectious diseases of humans: dynamics and control

Modeling infectious diseases in humans and animals

Dynamics and epidemiological impact of microparasites

Measles and rubella misdiagnosed in infants as Exanthem subitum

A contribution to the mathematical theory of epidemics

Measles periodicity and community size

Measles in England and Wales-I: an analysis of factors underlying seasonal patterns

Mathematical models for infectious disease statistics

An age-structured model of pre-and post-vaccination measles transmission

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics

Viral infections of humans: epidemiology and control

The epidemiology of varicella zoster virus-infections -a mathematical model

Natural immune boosting in pertussis dynamics: the potential for long-term vaccine failure

Measuring the performance of vaccination programs using cross-sectional surveys: a likelihood framework and retrospective analysis

Herd-immunity -history, theory, practice

History of International Public Health No. 6. Geneva: World Health Organization

Cities and villages: infection hierarchies in a measles metapopulation

Seasonal infl uences on pertussis

Stochastic amplifi cation in epidemics

Deterministic and stochastic models for recurrent epidemics

A simple model for complex dynamical transitions in epidemics

Measles as a case-study in nonlinear forecasting and chaos

The dynamics of measles in sub-Saharan Africa

Measles in developing countries. Part II. The predicted impact of mass vaccination

Contact surface models for infectious diseases: estimation from serologic survey data

Estimation of infectious disease parameters from serological survey data: the impact of regular epidemics

Using data on social contacts to estimate age-specifi c transmission parameters for respiratoryspread infectious agents

The prevaccination epidemiology of measles, mumps and rubella in Europe: implications for modelling studies

Interpretation of serological surveillance data for measles using mathematicalmodels -implications for vaccine strategy

A catalytic model of infection for measles

Social contacts and mixing patterns relevant to the spread of infectious diseases

Contact network structure explains the changing epidemiology of pertussis

Episodic outbreaks bias estimates of age-specifi c force of infection: a corrected method using measles as an example

The shifting demographic landscape of pandemic infl uenza

The history and medical consequences of rubella

Impact of birth rate, seasonality and transmission rate on minimum levels of coverage needed for rubella vaccination

The critical community size for measles in the U.S

Measles endemicity in insular populations: critical community size and its evolutionary implication

Hazards, spatial transmission and timing of outbreaks in epidemic metapopulations

Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics

Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes

Air travel and the spread of infl uenza: Important caveats

Demographic variability, vaccination, and the spatiotemporal dynamics of rotavirus epidemics

Meta) population dynamics of infectious diseases

Population biology of microparasitic infections

Respiratory syncytial virus: the virus, the disease and the immune response

Viral shapeshifting: norovirus evasion of the human immune system

Positive selection on the H3 hemagglutinin gene of human infl uenza virus A

Dynamical resonance can account for seasonality of infl uenza epidemics

Protective effect of natural rotavirus infection in an Indian birth cohort

Rotavirus infections in infants as protection against subsequent infections

The population dynamical implications of covert infections in hostmicroparasite interactions

Mass vaccination to control chickenpox: the infl uence of zoster

Dengue: a continuing global threat

Unifying the epidemiological and evolutionary dynamics of pathogens

Infl uenza escapes immunity along neutral networks

A prospective seroepidemiologic study on dengue in children four to nine years of age in Yogyakarta, Indonesia I. Studies in 1995-1996

Evidence that maternal dengue antibodies are important in the development of dengue hemorrhagic fever in infants

Risk factors in dengue shock syndrome: a prospective epidemiologic study in Rayong, Thailand. I. The 1980 outbreak

Monoclonal antibody-mediated enhancement of dengue virus infection in vitro and in vivo and strategies for prevention

Antibody, macrophages, dengue virus infection, shock, and hemorrhage: a pathogenetic cascade

Dynamic models of infectiousdiseases as regulators of population sizes

Immunological serotype interactions and their effect on the epidemiological pattern of dengue

The effect of antibodydependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens

Research on dengue during World War II

Cross-protective immunity can account for the alternating epidemic pattern of dengue virus serotypes circulating in Bangkok

Ecological and immunological determinants of dengue epidemics

Decreases in dengue transmission may act to increase the incidence of dengue hemorrhagic fever

The evolution and emergence of RNA viruses

Heterogeneity of HIV-1 and HIV-2

Mapping the antigenic and genetic evolution of infl uenza virus

Molecular mechanisms of variation in infl uenza viruses

Hemagglutinin receptor binding avidity drives infl uenza A virus antigenic drift

Effects of glycosylation on the properties and functions of infl uenza virus hemagglutinin

N-linked glycosylation attenuates H3N2 infl uenza viruses

Dynamics of glycoprotein charge in the evolutionary history of human infl uenza

Changing selective pressure during antigenic changes in human infl uenza H3

Infl uenza vaccine: the challenge of antigenic drift

Epochal evolution shapes the phylodynamics of interpandemic infl uenza A (H3N2) in humans

Ecological and immunological determinants of infl uenza evolution

Evolutionary rate at the molecular level

The generation of infl uenza outbreaks by a network of host immune responses against a limited set of antigenic types

The causes and consequences of genetic variation in dengue virus

Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future

Parasite-host coevolution

Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis

The epidemiological consequences of the mechanical transmission of myxomatosis by mosquitoes

Modelling viral and immune system dynamics

The role of models in understanding CD8+ T-cell memory

Mathematical models of cytotoxic T-lymphocyte killing

The human immunodefi ciency virus: infectivity and mechanisms of pathogenesis

HIV-1 dynamics in vivo: virion clearance rate, infected cell lifespan, and viral generation time

Viral dynamics in human immunodefi ciency virus type 1 infection

Virus dynamics and drug therapy

Understanding the failure of CD8+ T-cell vaccination against simian/human immunodefi ciency virus

Hepatitis C virus infection

Modelling how ribavirin improves interferon response rates in hepatitis C virus infection

Effect of ribavirin on hepatitis C viral kinetics in patients treated with pegylated interferon

Hepatitis C viral dynamics in vivo and the antiviral effi cacy of interferon-alpha therapy

New kinetic models for the hepatitis C virus

The kinetics of hepatitis C virus infection

A review of mathematical models of infl uenza A infections within a host or cell culture: lessons learned and challenges ahead

Dynamics of infl uenza virus infection and pathology

Kinetics of infl uenza A virus infection in humans

Towards a quantitative understanding of the within-host dynamics of infl uenza A infections

Understanding the mechanisms and limitations of immune control of HIV

Clinical implications of hepatitis C viral kinetics

Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape

Planning for smallpox outbreaks

Strategies for mitigating an infl uenza pandemic

Quantifying the impact of immune escape on transmission dynamics of infl uenza

Assessing vaccination sentiments with online social media: implications for infectious disease dynamics and control

Use of unstructured eventbased reports for global infectious disease surveillance

Ultra-deep sequencing for the analysis of viral populations

Maximizing power in seroepidemiological studies through the use of the proportional odds model. Infl uenza Other Respi Viruses

The role of seroepidemiology in the comprehensive surveillance of vaccinepreventable diseases

Epidemiology of genital herpes simplex virus infection in developed countries

Serological evidence of subclinical transmission of the 2009 pandemic H1N1 infl uenza virus outside of Mexico

Cumulative impact of host and viral factors on HIV-1 viral-load control during early infection

Superspreading and the effect of individual variation on disease emergence

Understanding individual human mobility patterns

Explaining seasonal fl uctuations of measles in Niger using nighttime lights imagery

Time series modelling of childhood diseases: a dynamical systems approach