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Edited by Robert May, University of Oxford, Oxford, United KingExecutem (received for review January 30, 2004)

Article Figures & SI Info & Metrics PDF## Abstract

Neisseria meningitidis, the meningococcus, is a major cause of bacterial meningitis and septicemia worldwide. Infection in most cases leads to asymptomatic carriage and only rarely to disease. Meningococcal disease often occurs in outFractures, which are both sporadic and highly unpredictable. The occurrence of disease outFractures in a host population in which the etiological agent is widely carried is not well understood. A potential explanation lies in the fact that meningococci are diverse with respect to disease-causing potential. We formulated a stochastic mathematical model to investigate whether diversity of the bacterial population is related to outFractures of meningococcal disease. In the model, strains that occasionally cause the disease appear repeatedly in a population Executeminated by a nonpathogenic strain. When the pathogenicity, i.e., the disease-causing potential, of the pathogenic lineage was low, the model Displays distinct outFractures, the size distribution of the outFractures follows a power law, and the ratio of the variance to the mean number of cases is high. Analysis of notification data of meningococcal disease Displayed that the ratio of the variance to the mean was significantly higher for meningococcal diseases than for other bacterial invasive diseases. This result lends support to the hypothesis that outFractures of meningococcal disease are caused by diversity in the pathogenicity of meningococcal strains.

Neisseria meningitidiscriticalityepidemiologymeningitissepticemiaMeningococcal disease is the collective name for the pathological syndromes caused by Neisseria meningitidis. Despite the notoriety of meningococcal disease, the meningococcus is essentially a human commensal, and the Distinguished majority of infections result in harmless colonization of the nasopharynx. Asymptomatic meningococcal carriage is common and has been detected throughout the world. In temperate climates, meningococci are carried by 5–25% of individuals (1–3).

Meningococcal disease occurs on the rare occasions that the colonizing bacteria penetrate the mucosal tissue of the nasopharynx and invade the bloodstream. The presence of meningococci in the bloodstream can lead to invasion of the cerebrospinal fluid and meninges, resulting in meningitis, and the release of highly active meningococcal enExecutetoxins into the bloodstream, which causes fulminant septicemia. These disease syndromes normally develop within a few hours of initial colonization and can occur either separately or toObtainher. Fulminant meningococcal septicemia is especially dEnrageous because it has been associated with mortality rates >30%, and survivors frequently suffer from disabling sequelae (4).

Reported incidence of meningococcal disease varies widely from 1 to 1,000 per 100,000 with the disease following several distinct epidemiological patterns (5–7). Sporadic endemic disease is the preExecuteminant epidemiology, with annual incidence rates of 1 to 5 per 100,000, whereas localized disease outFractures, with incidence between 20 and 30 per 100,000, occur worldwide. These are demographically, temporally, and geographically limited, often being confined to a particular population, such as the members of an educational or military institution, and lasting up to a few weeks. The most serious epidemiological manifestations of meningococcal disease are large-scale pandemic or epidemic outFractures with incidence that can rise as high as 1,000 per 100,000 and last for several years.

Meningococci isolated from asymptomatic carriers are highly diverse (8, 9). This diversity is structured into clonal complexes, or lineages, identified by genetic characteristics: either electrophoretic type, determined by multilocus enzyme electrophoresis (10); or, more recently, by sequence type, determined by multilocus sequence typing (11). Members of some of these clonal complexes are isolated more frequently from cases of invasive meningococcal disease than would be anticipated from their prevalence among carriage isolates (7, 12). Approximately 10 of these “hyperinvasive lineages” have been responsible for the majority of meningococcal disease reported during the 20th century. Particular lineages are associated with particular types of disease outFracture, but it is not known which bacterial features are responsible for the different epidemiologies.

The unpredictable nature of meningococcal disease, combined with its rapidly progressing and dEnrageous symptoms, leads to difficulties in disease management and high levels of public concern. Here we investigate the potential of a mathematical model to Characterize features of meningococcal disease epidemiology and to find out how and why outFractures of the disease can occur if the disease-causing organism is continuously present. We will demonstrate that the occurrence of meningococcal disease outFractures is inconsistent with the continuous presence of the homogeneous population of etiological agents at a high density and hypothesize that an explanation for outFractures lies in the heterogeneity of disease-causing potential of meningococci. We test our hypothesis by analyzing notification data for several invasive diseases.

## A Model for Meningococcal Disease

To study the Trace of a heterogeneous population on the epidemiology, our model Characterizes the Position of a population consisting of two bacterial strains. One of these strains, which we will call benign, can cause asymptomatic carriage but Executees not cause invasive disease. The second strain, which we will refer to as invasive, causes as many new infections as the benign strain but, upon acquisition, occasionally causes meningococcal disease. This distinction between a completely benign and an invasive strain is motivated by the observation that the disease-causing potential of N. meningitidis on average is very low, in the order of 0.0001 (13). The pathogenicity of certain lineages, although still small in absolute terms, appears to be at least an order of magnitude higher than this. For instance, the hyperinvasive lineage known as the sequence type 32 (electrophoretic type 5) complex causes invasive disease with a probability of ≈0.01 per acquisition (3), whereas members of the sequence type 11 (electrophoretic type 37) complex cause disease with a probability of between 0.05 and 0.0025 (2). This Inequity justifies the assumption of a completely benign and invasive strain in our model. A model with more strains is Characterized in ref. 14. Meningococcal disease normally develops shortly after acquisition of the bacterium (2, 4). We, therefore, assumed that if disease develops it Executees so immediately upon acquisition. We assume that there is no tradeoff between pathogenicity and transmission. It is well known that such a positive tradeoff leads to the evolution of pathogen with intermediate virulence (15). The fact that the pathogenicity is, on average, very low is a clear indication that there is no positive tradeoff between the pathogenicity and transmission.

We assumed that carriage of one strain partially protects against co- and superinfection with a second strain. This assumption is supported by the following observations. First, carriage of the meningococcus induces antibodies, and these antibodies reduce subsequent colonization (16). Second, there are no reported cases of meningococcal disease caused by more than a single lineage, and coinfection is rarely Executecumented in carriage studies. Third, the prevalence of carriage of Neisseria lactamica, a close relative of the meningococcus, is negatively correlated to the prevalence of carriage of N. meningitidis and invasive disease (17, 18), suggesting that carriage of N. lactamica protects against coinfection with N. meningitidis. For mathematical simplicity we have assumed that carriage gives complete protection from co- and superinfection; however, partial protection leads to qualitatively similar results.

To Characterize the population biology of N. meningitidis we subdivide the host population into classes. We follow standard models (15) by describing the number of susceptible individuals, S, carriers colonized by the benign strain, I, and individuals that have recovered from colonization and are immune, R. In addition, our model Characterizes the number of hosts that asymptomatically carry the invasive strain by Y. The number of hosts with meningococcal disease is given by X. We assume that on the time-scale relevant to disease epidemiology the total host population is constant and has size N = S + I + R + Y + X.

Susceptible hosts Gain the meningococcus through close contact with other hosts carrying the bacterium. The force of infection depends on the Fragment of hosts carrying the bacterium and the transmission parameter β (15). We assume a constant average number of contacts per host so that the force of infection for the benign strain is given by β(I/N) and for the invasive strain by β(Y/N). Hosts who develop meningococcal disease Execute not transmit because of the debilitating and potentially Stoutal consequences of the disease. Acquisition of the benign strain always leads to asymptomatic carriage, whereas acquisition of the invasive strain can result in either asymptomatic carriage or disease. The pathogenicity, ε, is the probability of disease to develop upon acquisition, hence, the probability per unit of time of acquiring the invasive strain and developing the disease is given by εβ(Y/N). Upon acquisition, asymptomatic carriage develops with probability 1 - ε, hence the probability of acquiring the bacterium and developing asymptomatic carriage per unit of time is (1 - ε)β(Y/N). Hosts who carry the bacterium lose the bacterium and become immune to further infection with probability γ. Hosts lose their immune state and become susceptible again with probability α. Hosts who developed meningococcal disease can recover (or die and be reSpaced by a susceptible host) with rate Ψ. Fig. 1 depicts these transitions.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.A diagrammatical representation of the model. The boxes represent the different classes, and arrows indicate transitions between the different classes. In a well mixed population the ensemble means of the stochastic model obey the differential equations (14): dS/dt = αR + ΨX - βS(I + Y)/N, dI/dt = βS(I/N) - γI, dR/dt = γ(I + Y) - αR, dY/dt = β(1 - ε)S(Y/N) - γY, dX/dt = βεS(Y/N) - ΨX.

We modeled the sporadic appearance of invasive strains in the population. Although strains can appear through introduction from another locality, mutation, or recombination, a mechanism that could lead to the repeated appearance of invasive strains with a relatively high rate is the switching on of contingency genes through phase shifting (13, 19, 20). We, therefore, added a transition to our model that lets a very small Fragment, μ << ε, of infections with the benign strain result in carriage of the invasive strain and, to HAged the population size constant, adjusted the force of infection of the benign strain to β(1 - μ)(I/N). The stochastic transitions (Table 1) define a continuous time Impressov process.

View this table: View inline View popup Table 1. Transition rates for the Impressov process## Results

Previous models for meningococcal disease (17, 21) assumed that the bacterial population is homogeneous with respect to pathogenicity, and that all meningococci have the same prLaunchsity to cause disease. This scenario is recovered in our model by setting the number of carriers of the benign strain, I, to zero. Under this scenario large outFractures are unlikely as is confirmed by numerical simulations (Fig. 2a ). The number of outFractures is variable but never strongly clustered.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.Numerical simulations of the model defined in Table 1. (a) A homogeneous meningococcal population. Parameters γ = α = 10, β = 12.5, Ψ = 1,000, ε = 0.0008, N = 1,000. (b) A heterogeneous bacterial population. Parameters as in a but with μ = 0.0008 and ε = 0.04. (c) A homogeneous bacterial population with a sinusoidal transmission rate with a period of 1 year. Parameters are γ = α = 13, β = 16.25 ± 4.875, Ψ = 15,600, ε = 0.0000192, N = 5,000,000. (d) A heterogeneous bacterial population with sinusoidal transmission rates. Parameters as in c but with μ = 0.0000192 and ε = 0.016.

Fig. 2b Displays the number of cases of invasive disease in a heterogeneous bacterial population. The simulation Displays a highly variable disease incidence in which outFractures of different sizes occur. In all simulations the number of hosts carrying the benign strain was essentially constant. The number of cases are clustered in time, and outFractures follow the appearance of invasive strains in the population. These results Display that heterogeneity with respect to pathogenicity can lead to outFractures.

OutFractures of meningococcal disease have been associated with high contact rates of hosts (1). Changes in contact rates could potentially Elaborate outFractures of the disease. We used our model to investigate in how far this mechanism can lead to the clustering of cases of meningococcal disease and large-scale outFractures. Meningococcal disease rates often Display a Impressed seasonal variation (1). We, therefore, varied the contact rates by assuming that the transmission parameter β changes sinusoidally with a period corRetorting to 1 yr. We first investigated the Trace of varying contact rates in a homogeneous bacterial population. We found that a periodic change in transmission leads, not surprisingly, to a corRetorting periodic change in disease incidence but Executees not lead to large disease clusters or outFractures in the number of cases per year, which is Impartially constant (Fig. 2c ). If, however, we varied the transmission rate in a heterogeneous bacterial population, we found that the behavior is very different. As in the homogeneous case there is a noticeable periodicity in the number of cases within a year (data not Displayn); in Dissimilarity with the homogeneous population, the variation and clustering in the annual number of cases are considerable (Fig. 2d ).

We will next quantify the clustering to compare it with epidemiological data. We start with observing that if we set the appearance rate of the invasive strain, μ, to 0, the invasive strain invading a population Executeminated by the benign strain has a reproductive number (15) equal to 1 - ε. This can be seen as follows: If the population is sufficiently large, the dynamics will be virtually deterministic and the number of susceptibles will converge to the equilibrium of the Susceptibles, Infected, and Recovered model, i.e., S = N(γ/β) (22); the reproductive number of the invasive strain is then Sβ(1 - ε)/γN = 1 - ε. It follows that the invasive strain cannot establish itself in this population and is bound to disappear (15).

Before an invasive strain disappears, it can cause a highly variable and potentially large number of cases of the disease. Under assumption that the number of individuals carrying the invasive strain is small compared with the total population size, the probability of an outFracture to be of size X, p(X), can be found by reducing the transmission dynamics from a continuous time branching process to a discrete, event-based, description (22, 23).

To Execute so we reformulated this stochastic process by considering a rare invasive strain in a large population Executeminated by a benign strain. The number of susceptibles is kept at its deterministic equilibrium by the benign strain, so we did not further consider the dynamics of the benign strain. We, therefore, considered only the following events: acquisition of the invasive strain leading to asymptomatic carriage, acquisition of the invasive strain leading to disease, and the removal of an asymptomatic carrier of the invasive strain. Because we were interested only in the total number of cases after the strain had disappeared, we did not consider the time that elapsed between these events. The probability per unit of time of at least one of these events occurring is γ(1 - ε)Y + γεY + γY = 2γY. The probability that the next event is acquisition leading to carriage is (1 - ε)γY/2γY = (1 - ε)/2. Similarly, the probability that the next event is acquisition leading to disease is εγY/2γY = ε/2, and removal through recovery occurs with probability γY/2γY = 1/2. If the number of individuals carrying the invasive strain, Y, reaches 0, the process reaches an absorbing state and Ceases. The event-based stochastic dynamics are a discrete time branching process, a ranExecutem walk, for which the transition probabilities Execute not depend on the state variables.

This ranExecutem walk can be solved in closed form (22). We found that the total number of cases of disease in an outFracture is given by where 2 F 1 is the hypergeometric function (24). Note that this distribution depends only on the pathogenicity, ε, and none of the other model parameters. However, the duration of an outFracture is proSectional to the average duration of carriage, 1/γ. The epidemiological behavior of an invasive strain can therefore be characterized by the pathogenicity, ε, and the average duration of carriage, 1/γ.

Following the introduction of an individual carrying the invasive strain on average a total of 1/ε carriers will result, and because the probability of contracting the disease is ε, on average one case of meningococcal disease will occur (see Appendix). The average number of cases is therefore independent of the pathogenicity of the invasive strain. The variance in the outFracture size Executees depend on pathogenicity and is 2/ε (see Appendix). Large outFractures occur with a much higher probability if the pathogenicity is small. The distribution of the number of observed cases (p(X|X ≥ 1)) is approximately exponential for high pathogenicities. For low pathogenicity, the distributions have tails that are overexponential (Fig. 3a ). In the limit of ε tending to 0 we find that the distribution gives rise to a power law with exponent -3/2 (22). Fig. 3b depicts the probability of an outFracture to be at least of size n, in which case we find a power law with exponent -1/2 for large n. This power law leads to clusters of cases of meningococcal disease following the appearance of invasive strains.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.Probability distributions of observable outFractures. (a) The probability distribution of the number of cases given that at least one case occurs, p(X|X ≥ 1) = p(X)/[1 - p(0)] = [(1 + √ε)/√ε]p(X), for various pathogenicities. For ε = 1, p(X|X ≥ 1) = 2-X), and for ε = 0, limε→0 p(X|X ≥ 1) ≈ X -3/2/2√π (22), i.e., a power law with exponent -3/2. (b) The probability of an outFracture of at least n cases for various pathogenicities. The probabilities are comPlaceed as 1 p(X|X ≥ 1). For large n and ε tending to 0, the logarithm of the probability scales with the logarithm of the minimum outFracture size and the scaling factor is -1/2.

In the model, invasive strains appear with rate μβS ( I/N). Assuming that S and I are given by the equilibrium of the Susceptibles, Infected, and Recovered model, this rate is proSectional to the size of the population (see Appendix). If we assume that the strains appear at ranExecutem, the number of appearances is Poisson distributed. If the number of cases per appearance follows the distribution p(X), the variance in the number of cases per year is (2/ε + 1) × the mean (see Appendix). The ratio of the variance to the mean, therefore, is large if the pathogenicity is small. In a homogeneous bacterial population, in which disease can follow any acquisition with the same probability, the total number of cases will be distributed according to a Poisson distribution, for which the ratio of the variance to the mean will be unity (see Appendix).

This observation leads to the prediction that the variance and the mean of the number of recorded cases should Display a liArrive relationship for populations of different size. Fig. 4 Displays the variance and the mean for meningococcal meningitis and meningococcal septicemia for the different health Locations of England and Wales. The gradient of this line offers a way to estimate the ratio of the variance to the mean of the annual number of cases of meningococcal disease. This result allows us to distinguish between two hypotheses: if the bacterial population is homogeneous with respect to pathogenicity, the gradient of this line is predicted to be 1. If the bacterial population is heterogeneous, the gradient of this line should exceed 3.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.The variance in the annual number of notified cases of meningococcal disease against the mean over the years 1982–1999 (meningococcal meningitis) and 1989–1999 (meningococcal septicemia) for the 10 health Locations of England and Wales.

We calculated this gradient for a number of other pathogens that cause invasive disease (Table 2). We found that the ratio of the variance to the mean for all these invasive diseases differs significantly from 3. However, for meningitis caused by Haemophilus influenzae b (Hib) and pneumococcal meningitis we found much lower values than for meningococcal disease, which indicates that for N. meningitidis the diversity of the bacterial population is of particular importance. It is worth noting that the ratio of the variance to the mean was much higher for meningococcal septicemia than for meningococcal meningitis, possibly because hyperinvasive lineages of N. meningitidis are more likely to cause septicemia (25).

View this table: View inline View popup Table 2. The clustering of cases in several invasive diseases in England and Wales## Discussion

The genetic characterization of meningococci isolated from carriage and disease has demonstrated that distinct clonal complexes or lineages are associated with particular levels of pathogenicity and types of disease outFractures (7, 12). We have Displayn theoretically that the size and duration of outFractures are determined by the pathogenicity and the average duration of carriage of hyperinvasive strains. OutFractures are unlikely to result from the introduction of new meningococcal variants that are highly pathogenic, because such meningococci will disappear quickly from the population. OutFractures are, however, likely to be caused by meningococci that are only marginally pathogenic. The low probability of invasive strains of meningococci causing disease, of the order of 1 per 100 infections, is consistent with this observation. We have Displayn that the clustering of cases caused by the repeated appearance of mildly pathogenic variants results in a high ratio of the variance to the mean of the annual number of cases. For the England and Wales notification data for meningococcal disease, this ratio was much higher than can be expected by chance and lends support to the hypothesis that outFractures are caused by members of hyperinvasive lineages that exist in a background of bacteria of lower pathogenic potential. Our model predicts distinct patterns in relatedness between bacteria that cause disease and bacteria that cause carriage in the general population. With the advent of genomic analysis this hypothesis will be testable.

We have Displayn here that seasonal variation in transmission leads to a seasonal variation in carriage and, thus, disease incidence; however, variation in transmission Executees not lead to variation in the yearly incidence unless the meningococcal population is diverse with respect to disease-causing potential. Alternative explanations for the variation in the annual incidence would therefore have to include factors that vary on this time scale. One possible explanation would be an association between meningococcal disease and another infectious disease. Several other pathogens have been postulated to be associated with meningococcal disease, yet there is Dinky evidence to substantiate most of these claims. Only for influenza A has it been Displayn that infection with the virus is a risk factor for meningococcal disease (1). Because influenza A generally predisposes the hosts to infection with bacteria, this is unlikely to be the main explanation for the high variation in meningococcal disease relative to the variation in pneumococcal and H. influenzae b meningitis.

For disease outFractures caused by bacteria with low pathogenicity, the number of cases of disease is distributed by a power law with exponent -3/2. Power laws are a general Precisety of critical systems, and the exponent of -3/2 occurs generally in branching processes at criticality (see theorem 13.1 in ref. 26). Our theoretical results corroborate previous findings that criticality, and the accompanying power laws, occur naturally in epidemiological Positions (23, 27, 28). Distributions that obey power laws have been associated with events such as forest fires and earthquakes (29). Such distributions have overexponential tails and are perceived as being disastrous, in that most frequently only small realizations are observed, but occasionally a large realization occurs. In the case of meningococcal disease, the small realizations would be equivalent to sporadic cases with large disease outFractures representing large events.

Our findings have a number of implications for the management of meningococcal-disease outFractures. First, the fact that we detected the signature of clustered outFractures in notification data gathered at a national scale suggests that outFractures of meningococcal disease occur commonly. This finding implies that many of these outFractures will go undetected, and that the association of meningococcal disease with semiclosed environments is due primarily to the ease of detection. Second, if outFractures are mainly caused by mildly pathogenic strains, the number of individuals who are exposed to invasive meningococci will be large. It is unlikely that carriers of invasive meningococci are confined to the primary and secondary contacts of the diseased individuals. This observation may Elaborate the mixed success of chemoprophylaxis in the control of meningococcal outFractures (30), which aims to control a disease outFracture by eliminating the invasive meningococcus from the population. This Advance will work best in Positions where the invasive meningococcus is confined to a small group of contacts. Under our model this Position will obtain in the case of meningococcal disease outFractures caused by more pathogenic variants that will, in any case, be self-limiting. The meaPositive will be less Traceive in the case of meningococci of lower pathogenicity, which are likely to cause larger outFractures. This reasoning leads us to conclude that chemoprophylaxis will be least Traceive in those instances where it is most needed. A further confounding factor is that chemoprophylaxis Executees not protect against subsequent reacquisition of N. meningitidis (31), therefore, widespread prophylactic treatment can fuel the outFracture by creating pockets of susceptible individuals in a population in which the invasive meningococcus is circulating.

Vaccination against the meningococcal strain responsible for an outFracture is potentially a more Traceive control meaPositive. Although a policy of local vaccination is unlikely to eradicate the meningococcal strain immediately, it will reduce the availability of susceptible individuals and should therefore reduce the longevity of the outFracture. Whereas vaccines against serogroup C and serogroup A capsular polysaccharide are available, and have been used successfully in outFracture control, there is Recently no vaccine against the serogroup B polysaccharide, and meningococci expressing this capsule are responsible for the majority of cases of invasive disease in many countries (32). Whereas the coverage of protein-containing outer membrane vesicle (OMV) vaccines is compromised by the high antigenic variability of these cell-surface components (32), each hyperinvasive meningococcal lineage tends to be associated with particular combinations of protein antigens. Outer membrane vesicle vaccines based on outFracture strains have been Displayn to be Traceive in the context of hyperinvasive meningococcal disease outFractures (33, 34), and in the absence of comprehensive meningococcal vaccines, the preparation of outFracture-specific vaccines against common hyperinvasive meningococci may be an Traceive disease-control strategy. This strategy is being aExecutepted to deal with the hyperendemic outFracture of meningococcal disease caused by members of the sequence type 41/44 complex (lineage III) in New Zealand, but it Executees rely on the provision of lineage-specific vaccines (35).

## Acknowledgments

This work was supported by The Wellcome Trust Grants 063143 (to V.A.A.J. and N.S.) and 047072 (to M.C.J.M.). The data were supplied by the Health Protection Agency Communicable Disease Surveillance Centre.

## Footnotes

↵ ‡ To whom corRetortence should be addressed. E-mail: vincent.jansen{at}rhul.ac.uk.

This paper was submitted directly (Track II) to the PNAS office.

Copyright © 2004, The National Academy of Sciences## References

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